Second order symmetry operators

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Second order symmetry operators

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2014 Class. Quantum Grav. 31 135015

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Classical and Quantum Gravity

Class. Quantum Grav. 31 (2014) 135015 (38pp) doi:10.1088/0264-9381/31/13/135015

Second order symmetry operators

Lars Andersson1, Thomas Backdahl2 and Pieter Blue2

1 Albert Einstein Institute, Am Muhlenberg 1, D-14476 Potsdam, Germany2 The University of Edinburgh, James Clerk Maxwell Building, Mayfield Road,Edinburgh, EH9 3JZ, UK

E-mail: [email protected], [email protected] and [email protected]

Received 12 March 2014Accepted for publication 26 March 2014Published 17 June 2014

AbstractUsing systematic calculations in spinor language, we obtain simple descriptionsof the second order symmetry operators for the conformal wave equation, theDirac–Weyl equation and the Maxwell equation on a curved four-dimensionalLorentzian manifold. The conditions for existence of symmetry operators forthe different equations are seen to be related. Computer algebra tools have beendeveloped and used to systematically reduce the equations to a form whichallows geometrical interpretation.

Keywords: maxwell equation, Dirac–Weyl equation, massless fields, symmetryoperators, Killing spinorsPACS numbers: 02.30.Jr, 02.70.Wz, 04.20.−q

S Online supplementary data available from stacks.iop.org/CQG/31/135015/mmedia

1. Introduction

The discovery by Carter [1] of a fourth constant of the motion for the geodesic equationsin the Kerr black hole spacetime, allowing the geodesic equations to be integrated, togetherwith the subsequent discovery by Teukolsky, Chandrasekhar and others of the separability ofthe spin-s equations for all half-integer spins up to s = 2 (which corresponds to the case oflinearized Einstein equations) in the Kerr geometry, provides an essential tool for the analysisof fields in the Kerr geometry. The geometric fact behind the existence of Carter’s constantis, as shown by Walker and Penrose [2], the existence of a Killing tensor. A Killing tensor isa symmetric tensor Kab = K(ab), satisfying the equation ∇(aKbc) = 0. This condition impliesthat the quantity K = Kabγ

aγ b is constant along affinely parametrized geodesics. In particular,viewed as a function on phase space, K Poisson commutes with the Hamiltonian generatingthe geodesic flow, H = γ aγa.

Carter further showed that in a Ricci flat spacetime with a Killing tensor Kab, theoperator K = ∇aKab∇b, which may be viewed as the ‘quantization’ of K, commutes with

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Class. Quantum Grav. 31 (2014) 135015 L Andersson et al

the d’Alembertian H = ∇a∇a, which in turn is the ‘quantization’ of H, cf [3]. In particular,the operator K is a symmetry operator for the wave equation Hφ = 0, in the sense that it mapssolutions to solutions. The properties of separability, and existence of symmetry operators, forpartial differential equations are closely related [4]. In fact, specializing to the Kerr geometry,the symmetry operator found by Carter may be viewed as the spin-0 case of the symmetryoperators for the higher spin fields as manifested in the Teukolsky system, see e.g. [5, 6].

In this paper we give necessary and sufficient conditions for the existence of secondorder symmetry operators, for massless test fields of spin 0, 1/2, 1, on a globally hyperbolicLorentzian spacetime of dimension 4. (As explained in section 2.4, the global hyperbolicitycondition can be relaxed.) In each case, the conditions are the existence of a conformal Killingtensor or Killing spinor, and certain auxiliary conditions relating the Weyl curvature and theKilling tensor or spinor. We are particularly interested in symmetry operators for the spin-1or Maxwell equation. In this case, we give a single auxiliary condition, which is substantiallymore transparent than the collection previously given in [7]. For the massless spin-1/2 or Dirac–Weyl equation, our result on second order symmetry operators represents a simplification of theconditions given by McLenaghan et al [8] for the existence of symmetry operators of order two.The conditions we find for spins 1/2 and 1 are closely related to the condition found recentlyfor the spin-0 case for the conformal wave equation by Michel et al [9], cf theorem 3 below.

A major motivation for the work in this paper is provided by the application by two of theauthors [10] of the Carter symmetry operator for the wave equation in the Kerr spacetime, toprove an integrated energy estimate and boundedness for solutions of the wave equation. Themethod used is a generalization of the vector fields method [11] to allow not only Killing vectorsymmetries but symmetry operators of higher order. In order to apply such methods to fieldswith non-zero spin, such as the Maxwell field, it is desirable to have a clear understanding ofthe conditions for the existence of symmetry operators and their structure. This serves as oneof the main motivations for the results presented in this paper, which give simple necessaryand sufficient conditions for the existence of symmetry operators for the Maxwell equationsin a four-dimensional Lorentzian spacetime.

The energies constructed from higher order symmetry operators correspond to conservedcurrents which are not generated by contracting the stress energy tensor with a conformalKilling vector. Such conserved currents are known to exist e.g. for the Maxwell equation,as well as fields with higher spin on Minkowski space, see [12] and references therein. In asubsequent paper [13] we shall present a detailed study of conserved currents up to secondorder for the Maxwell field.

We will assume that all objects are smooth, we work in signature (+,−,−,−), and we usethe 2-spinor formalism, following the conventions and notation of [14, 15]. For a translation tothe Dirac 4-spinor we refer to [13, p 221]. Recall that �/24 is the scalar curvature, �ABA′B′ theRicci spinor, and �ABCD the Weyl spinor. Even though several results are independent of theexistence of a spin structure, we will for simplicity assume that the spacetime is spin. The 2-spinor formalism allows one to efficiently decompose spinor expressions into irreducible parts.All irreducible parts of a spinor are totally symmetric spinors formed by taking traces of thespinor and symmetrizing all free indices. Making use of these facts, any spinor expression canbe decomposed in terms of symmetric spinors and spin metrics. This procedure is describedin detail in section 3.3 in [14] and in particular by proposition 3.3.54.

This decomposition has been implemented in the package SymManipulator [16]by the second author. SymManipulator is part of the xAct tensor algebra package[17] for Mathematica. The package SymManipulator includes many canonicalization andsimplification steps to make the resulting expressions compact enough and the calculationsrapid enough so that fairly large problems can be handled. A mathematica 9 notebook

2


file containing the main calculations for this paper is available as supplementary data atstacks.iop.org/CQG/31/135015/mmedia (see also: http://hdl.handle.net/10283/541).

We shall in this paper consider only massless spin-s test fields. For the spin-0 case thefield equation is the conformal wave equation

(∇a∇a + 4�)φ = 0, (1.1)

for a scalar field φ, while for non-zero spin the field is a symmetric spinor φA···F of valence(2s, 0) satisfying the equation

∇AA′φA···F = 0. (1.2)

In this paper we shall restrict our considerations to spins 0, 1/2, 1. For s � 3/2, equation (1.2)implies algebraic consistency conditions, which strongly restrict the space of solutions in thepresence of non-vanishing Weyl curvature. Note however that there are consistent equationsfor fields of higher spin, see [14, section 5.8] for discussion.

Recall that a Killing spinor of valence (k, l) is a symmetric spinor LA1···AkA′

1···A′l ,

∇(A1(A′

1 LA2···Ak+1 )A′

2···A′l+1) = 0. (1.3)

A valence (1, 1) Killing spinor is simply a conformal Killing vector, while a valence (2, 0)

Killing spinor is equivalent to a conformal Killing–Yano 2-form. On the other hand, a Killingspinor of valence (2, 2) is simply a trace-less symmetric conformal Killing tensor. It isimportant to note that (1.1), (1.2) and (1.3) are conformally invariant if φ and φA···F are givenconformal weight −1, and LA1···AkA′

1···A′l is given conformal weight 0. See [14, sections 5.7 and

6.7] for details.Recall that a symmetry operator for a system Hϕ = 0, is a linear partial differential

operator K such that HKϕ = 0 for all ϕ such that Hϕ = 0. We say that two operators K1

and K2 are equivalent if K1 − K2 = FH for some differential operator F . We are interestedonly in non-trivial symmetry operators, i.e. operators which are not equivalent to the trivialoperator 0. For simplicity, we will only consider equivalence classes of symmetry operators.

To state our main results, we need two auxiliary conditions.

Definition 1. Let LABA′B′

be a Killing spinor of valence (2, 2).

(A0) LABA′B′

satisfies auxiliary condition (A0) if there is a function Q such that

∇AA′Q = 13�ABCD∇ (B|B′|LCD)

A′B′ + 13 �A′B′C′D′∇B(B′

LABC′D′)

+LBCA′ B

′∇(AC′

�BC)B′C′ + LABB′C′∇C

(A′�|BC|B′C′). (1.4)

(A1) LABA′B′

satisfies auxiliary condition (A1) if there is a vector field PAA′

such that

∇(A(A′

PB)B′) = LCDA′B′

�ABCD − LABC′D′

�A′B′C′D′ . (1.5)

Remark 2. Under conformal transformations such that LABA′B′, PAA′

and Q are given conformalweight 0, the equations (1.4) and (1.5) are conformally invariant.

We start by recalling the result of Michel et al [9] for the spin-0 conformal wave equation,which we state here in the case of a Lorentzian spacetime of dimension 4.

Theorem 3 ([8, theorem 4.8]). Consider the conformal wave equation

(∇a∇a + 4�)φ = 0 (1.6)

in a four-dimensional Lorentzian spacetime. There is a non-trivial second order symmetryoperator for (1.6) if and only if there is a non-zero Killing spinor of valence (2, 2) satisfyingcondition (A0) of definition 1.

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Previous work on the conformal wave equation was done by [18], see also Kress [19],see also [20]. Symmetry operators of general order for the Laplace–Beltrami operator in theconformally flat case have been analyzed by Eastwood [21].

Next we consider fields with spins 1/2 and 1. The massless spin-1/2 equations are

∇AA′φA = 0, (1.7a)

and its complex conjugate form

∇AA′χA′ = 0, (1.7b)

which we shall refer to as the left and right Dirac–Weyl equations3. Analogously with theterminology used by Kalnins et al [7] for the spin-1 case, we call a symmetry operatorφA �→ λA, which takes a solution of the left equation to a solution of the left equation asymmetry operator of the first kind, while an operator φA �→ χA′ which takes a solution of theleft equation to a solution of the right equation a symmetry operator of the second kind.

If one considers symmetry operators in the Dirac 4-spinor notation, a 4-spinor wouldcorrespond to a pair of 2-spinors (φA, ϕA′ ). Therefore a symmetry operator (φA, ϕA′ ) �→(λA, χA′ ) for a 4-spinor is formed by a combination of symmetry operators of first φA �→ λA,and second φA �→ χA′ kind, together with complex conjugate versions of first ϕA′ �→ χA′ , andsecond ϕA′ �→ λA kind symmetry operators.

Theorem 4. Consider the Dirac–Weyl equations (1.1) in a Lorentzian spacetime ofdimension 4.

(i) There is a non-trivial second order symmetry operator of the first kind for the Dirac–Weylequation if and only if there is a non-zero Killing spinor of valence (2, 2) satisfyingauxiliary conditions (A0) and (A1) of definition 1.

(ii) There is a non-trivial second order symmetry operator of the second kind for the Dirac–Weyl equation if and only if there is a non-zero Killing spinor LABC

A′of valence (3, 1),

such that the auxiliary condition

0 = 34�ABCD∇FA′

LCDFA′ + 5

6�BCDF∇(A

A′LCDF )A′ + 5

6�ACDF∇(B

A′LCDF )A′

− 35 LB

CDA′∇(AB′�CD)A′B′ − 3

5 LACDA′∇(B

B′�CD)A′B′ + 4

3 LCDFA′∇(A|A′|�BCDF ) (1.8)

is satisfied.

Remark 5.

(i) Under conformal transformations such that LABCA′ = LABCA′, the equation (1.8) is

conformally invariant.(ii) We remark that the auxiliary condition (A0), appears both in theorem 4, and for the

conformal wave equation in theorem 3.

In previous work, Benn and Kress [22] showed that a first order symmetry operator ofthe second kind for the Dirac equation exists exactly when there is a valence (2, 0) Killingspinor. See also Carter and McLenaghan [23] and Durand et al [24] for earlier work. Theconditions for the existence of a second order symmetry operator for the Dirac–Weyl equationsin a general spacetime were considered in [8], see also [25]. The conditions derived hererepresent a simplification of the conditions found in [8]. Further, we mention that symmetry

3 The use of the terms left and right is explained by noting that spinors of valence (k, 0) represent left-handedparticles, while spinors of valence (0, k) represent right-handed particles, cf [14, section 5.7]. The Dirac equation isthe equation for massive, charged spin-1/2 fields, and couples the left- and right-handed parts of the field, see [14,section 4.4]. We shall not consider the symmetry operators for the Dirac equation here.

4


operators of general order for the Dirac operator on Minkowski space have been analyzed byMichel [26].

For the spin-1 case, we similarly have the left and right Maxwell equations

∇BA′φAB = 0, (1.9a)

∇AB′χA′B′ = 0. (1.9b)

The left-handed and right-handed spinors φAB, χA′B′ represent an anti-self-dual and a self-dual 2-form, respectively. Each equation in (1.9) is thus equivalent to a real Maxwell equation,cf [14, section 3.4]. Analogously to the spin-1/2 case, we consider second order symmetryoperators of the first and second kind.

Theorem 6. Consider the Maxwell equations (1.1) in a Lorentzian spacetime of dimension 4.

(i) There is a non-trivial second order symmetry operator of the first kind for the Maxwellequation if and only if there is a non-zero Killing spinor of valence (2, 2) such that theauxiliary condition (A1) of definition 1 is satisfied.

(i) There is a non-trivial second order symmetry operator of the second kind for the Maxwellequation if and only if there is a non-zero Killing spinor LABCD of valence (4, 0).

Note that no auxiliary condition is needed in point (ii) of theorem 6. The conditions forthe existence of second order symmetry operators for the Maxwell equations have been givenin previous work by Kalnins et al [7], see also [27], following earlier work by Kalnins et al[5], see also [19]. In [7], the conditions for a second order symmetry operator of the secondkind were analyzed completely, and agree with the condition given in point (ii) of theorem 6.However, the conditions for a second order symmetry operator of the first kind stated thereconsist of a set of five equations, of a not particularly transparent nature. The result given herein point (i) of theorem 6 provides a substantial simplification and clarification of this previousresult.

The necessary and sufficient conditions given in theorems 3, 4, 6 involve the existenceof a Killing spinor and auxiliary conditions. The following result gives examples of Killingspinors for which the auxiliary conditions (A0), (A1) and equation (1.8) are satisfied.

Proposition 7. Let ξAA′and ζ AA′

be (not necessarily distinct) conformal Killing vectors andlet κAB be a Killing spinor of valence (2, 0).

(i) The symmetric spinor ξ(A(A′

ζB)B′) is a Killing spinor of valence (2, 2), which admits

solutions to the auxiliary conditions (A0) and (A1).(ii) The symmetric spinor κABκA′B′ is also a Killing spinor of valence (2, 2), which admits

solutions to the auxiliary conditions (A0) and (A1).(iii) The spinor κ(ABξC)

C′is a Killing spinor of valence (3, 1), which satisfies auxiliary equation

(1.8).(iv) The spinor κ(ABκCD) is a Killing spinor of valence (4, 0).

The point (iv) is immediately clear. The other parts will be proven in section 6.We now consider the following condition

0 = �(ABCF�D)FA′B′ , (1.10)

relating the Ricci curvature �ABA′B′ and the Weyl curvature �ABCD. A spacetime where (1.10)holds will be said to satisfy the aligned matter condition. In particular this holds in vacuumand in the Kerr–Newman class of spacetimes. Under the aligned matter condition we can showthat the converse of proposition 7 part (iv) is true. The following theorem will be proved insection 6.5.

5


Theorem 8. If the aligned matter condition (1.10) is satisfied, �ABCD �= 0 and LABCD is avalence (4, 0) Killing spinor, then there is a valence (2, 0) Killing spinor κAB such that

LABCD = κ(ABκCD). (1.11)

Remark 9. If �ABCD = 0, the valence (4, 0) Killing spinor will still factor but in terms ofvalence (1, 0) Killing spinors, which then can be combined into valence (2, 0) Killing spinors.However, the two factors might be distinct.

A calculation shows that if (1.10) holds, κAB is a valence (2, 0) Killing spinor, thenξAA′ = ∇BA′

κAB is a Killing vector field. Taking this fact into account, we have the following

corollary to the results stated above. It tells that generically one can generate a wide variety ofsymmetry operators from just a single valence (2, 0) Killing spinor.

Corollary 10. Consider the massless test fields of spins 0, 1/2 and 1 in a Lorentzian spacetimeof dimension 4. Assume that there is Killing spinor κAB (not identically zero) of valence (2, 0).Then there are non-trivial second order symmetry operators for the massless spin-s fieldequations for spins 0 and 1, as well as a non-trivial second order symmetry operator of thefirst kind for the massless spin-1/2 field.

If, in addition, the aligned matter condition (1.10) holds, and ξAA′ = ∇BA′κAB is not

identically zero, then there is also a non-trivial second order symmetry operators of thesecond kind for the massless spin-1/2 field.

We end this introduction by giving a simple form for symmetry operators for the Maxwellequation, generated from a Killing spinor of valence (2, 0).

Theorem 11. Let κAB be a Killing spinor of valence (2, 0) and let

AB ≡ − 2κ(ACφB)C. (1.12)

Define the potentials

AAA′ = κA′ B′∇BB′ A

B − 13 A

B∇BB′ κA′ B′, (1.13a)

BAA′ = κAB∇CA′ B

C + 13 A

B∇CA′κBC. (1.13b)

Assume that φAB is a solution to the Maxwell equation in a Lorentzian spacetime ofdimension 4. Let AAA′ , BAA′ be given by (1.13). Then

χAB = ∇(BA′

AA)A′ , (1.14a)

ωA′B′ = ∇B(A′B|B|B′ ) (1.14b)

are solutions to the left and right Maxwell equations, respectively.

The proof can be found in sections 7.3 and 7.5. The general form of the symmetryoperators for spins 0, 1/2 and 1 is discussed in detail below.

Remark 12. The symmetry operators of the Maxwell equation can in general be written inpotential form. See theorems 25 and 27.

The method used in this paper can also be used to show that the symmetry operators R-commute with the Dirac and Maxwell equations. Recall that an operator S is said to R-commutewith a linear PDE Lφ = 0 if there is an operator R such that LS = RL. Even providing aformula for the relevant R operators would require additional notation, so we have omittedthis result from this paper.

6


Overview of this paper

In section 2 we define the fundamental operators D,C ,C †,T obtained by projecting thecovariant derivative of a symmetric spinor on its irreducible parts. These operators areanalogues of the Stein–Weiss operators discussed in Riemannian geometry and play a centralrole in our analysis. We give the commutation properties of these operators, derive theintegrability conditions for Killing spinors, and end the section by discussing some aspectsof the methods used in the analysis. Section 3 gives the analysis of symmetry operators forthe conformal wave equation. The results here are given for completeness, and agree withthose in [9] for the case of a Lorentzian spacetime of dimension 4. The symmetry operatorsfor the Dirac–Weyl equation are discussed in section 4 and our results for the Maxwell caseare given in section 5. Special conditions under which the auxiliary conditions can be solvedis discussed in section 6. Finally, section 7 contains simplified expressions for the symmetryoperators for some of the cases discussed in section 6.

2. Preliminaries

2.1. Fundamental operators

Let Sk,l denote the vector bundle of symmetric spinors with k unprimed indices and l primedindices. We will call these spinors symmetric valence (k, l) spinors. Furthermore, let Sk,l

denote the space of smooth (C∞) sections of Sk,l .

Definition 13. For any ϕA1...AkA′

1...A′l ∈ Sk,l , we define the operators Dk,l : Sk,l → Sk−1,l−1,

Ck,l : Sk,l → Sk+1,l−1, C †k,l : Sk,l → Sk−1,l+1 and Tk,l : Sk,l → Sk+1,l+1 as

(Dk,lϕ)A1...Ak−1A′

1...A′l−1 ≡ ∇BB′

ϕA1...Ak−1BA′

1...A′l−1 B′ , (2.1a)

(Ck,lϕ)A1...Ak+1A′

1...A′l−1 ≡ ∇(A1

B′ϕA2...Ak+1)

A′1...A

′l−1 B′ , (2.1b)

(C †k,lϕ)A1...Ak−1

A′1...A

′l+1 ≡ ∇B(A′

1ϕA1...Ak−1BA′

2...A′l+1), (2.1c)

(Tk,lϕ)A1...Ak+1A′

1...A′l+1 ≡ ∇(A1

(A′1ϕA2...Ak+1)

A′2...A

′l+1 ). (2.1d)

Remark 14.

(i) These operators are all conformally covariant, but the conformal weight differs betweenthe operators. See [15, section 6.7] for details.

(ii) The left Dirac–Weyl and Maxwell equations can be written as (C †1,0φ)A′ = 0 and

(C †2,0φ)AA′ = 0 respectively. Similarly the right equations can be written in terms of

the C operator.

The operator Dk,l only makes sense when k � 1 and l � 1. Likewise Ck,l is defined onlyif l � 1 and C †

k,l only if k � 1. To make a clean presentation, we will use formulas whereinvalid operators appear for some choices of k and l. However, the operators will always bemultiplied with a factor that vanishes for these invalid choices of k and l. From the definitionit is clear that the complex conjugates of (Dk,lϕ), (Ck,lϕ), (C †

k,lϕ) and (Tk,lϕ) are (Dl,kϕ),(C †

l,kϕ), (Cl,kϕ) and (Tl,kϕ) respectively, with the appropriate indices.The main motivation for the introduction of these operators is the irreducible

decomposition of the covariant derivative of a symmetric spinor field.

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Lemma 15. For any ϕA1...AkA′

1...A′l ∈ Sk,l , we have the irreducible decomposition

∇A1A′

1ϕA2...Ak+1A′

2...A′l+1 = (Tk,lϕ)A1...Ak+1

A′1...A

′l+1 − l

l + 1εA′

1(A′2 (Ck,lϕ)A1...Ak+1

A′3...A

′l+1)

− k

k + 1εA1(A2 (C

†k,lϕ)A3...Ak+1)

A′1...A

′l+1

+ kl

(k + 1)(l + 1)εA1(A2 ε

A′1(A

′2 (Dk,lϕ)A3...Ak+1)

A′3...A

′l+1 ). (2.2)

Proof. It follows from in [14, proposition 3.3.54] that the irreducible decomposition musthave this form. The coefficients are then found by contracting indices and partially expandingthe symmetries. �

With this notation, the Bianchi system takes the form

(D2,2�)AA′ = − 3(T0,0�)AA′ , (2.3a)

(C †4,0�)ABCA′ = (C2,2�)ABCA′ . (2.3b)

In the rest of the paper we will use these equations every time the left-hand sides appearin the calculations.

With the definitions above, a Killing spinor of valence (k, l) is an element LA···F A′...F ′ ∈ker Tk,l , a conformal Killing vector is a Killing spinor of valence (1, 1), and a trace-lessconformal Killing tensor is a Killing spinor of valence (2, 2). We further introduce thefollowing operators, acting on a valence (2, 2) Killing spinor.

Definition 16. For LABA′B′ ∈ ker T2,2, define(

O(0)

2,2L)

AA′ ≡ 1

3�ABCD(C2,2L)BCDA′ + LBCA′B′(C2,2�)ABCB′

+ 13 �A′

B′C′D′ (C †2,2L)A

B′C′D′ + LABB′C′

(C †2,2�)B

A′B′C′ . (2.4a)

(O(1)

2,2L)

ABA′B′ ≡ LCDA′B′


�A′B′C′D′ . (2.4b)

The operators O(0)

2,2 and O(1)

2,2 are the right-hand sides of (1.4) and (1.5) in conditions (A0)and (A1) respectively. They will play an important role in the conditions for the existence ofsymmetry operators.

Given a conformal Killing vector ξAA′, we follow [28, equations (1.2) and (15)], see also

[12], and define a conformally weighted Lie derivative acting on a symmetric valance (2s, 0)

spinor field as follows.

Definition 17. For ξAA′ ∈ ker T1,1, and ϕA1...A2s ∈ S2s,0, we define

LξϕA1...A2s ≡ ξBB′∇BB′ϕA1...A2s + sϕB(A2...A2s∇A1 )B′ξBB′ + 1 − s

4ϕA1...A2s∇CC′

ξCC′ . (2.5)

This operator turns out to be important when we describe first order symmetry operators.See section 7.4 for further discussion.

2.2. Commutator relations

Let ϕA1...AkA′

1...A′l ∈ Sk,l and define the standard commutators

�AB ≡ ∇(A|A′|∇B)A′

and �A′B′ ≡ ∇A(A′∇AB′ ). (2.6)

8


Acting on spinors, these commutators can always be written in terms of curvature spinors asdescribed in [14, section 4.9].

Lemma 18. The operators D , C , C † and T satisfies the following commutator relations

(Dk+1,l−1Ck,lϕ)A1...AkA′

1...A′l−2 = k

k + 1(Ck−1,l−1Dk,lϕ)A1...Ak

A′1...A

′l−2

−�B′C′ϕA1...AkA′

1...A′l−2B′C′

, k � 0, l � 2, (2.7a)

(Dk−1,l+1C†k,lϕ)A1...Ak−2

A′1...A

′l = l

l + 1(C †

k−1,l−1Dk,lϕ)A1...Ak−2A′

1...A′l

−�BCϕA1...Ak−2BCA′

1...A′l , k � 2, l � 0, (2.7b)

(Ck+1,l+1Tk,lϕ)A1...Ak+2A′

1...A′l = l

l + 1(Tk+1,l−1Ck,lϕ)A1...Ak+2

A′1...A

′l

−�(A1A2ϕA3...Ak+2)A′

1...A′l , k � 0, l � 0, (2.7c)

(C †k+1,l+1Tk,lϕ)A1...Ak

A′1...A

′l+2 = k

k + 1(Tk−1,l+1C

†k,lϕ)A1...Ak

A′1...A

′l+2

−�(A′1A′

2ϕA1...AkA′

3...A′l+2), k � 0, l � 0, (2.7d)

(Dk+1,l+1Tk,lϕ)A1...AkA′

1...A′l = −

(1

k + 1+ 1

l + 1

)(Ck−1,l+1C

†k,lϕ)A1...Ak

A′1...A

′l

+ l(l + 2)

(l + 1)2(Tk−1,l−1Dk,lϕ)A1...Ak

A′1...A

′l − l + 2

l + 1�B

(A1ϕA2...Ak )BA′

1...A′l

− l

l + 1�B′(A′

1ϕA1...AkA′

2...A′l )B′ , k � 1, l � 0, (2.7e)


1...A′l = −

(1

k + 1+ 1

l + 1

)(C †

k+1,l−1Ck,lϕ)A1...AkA′

1...A′l

+ k(k + 2)

(k + 1)2(Tk−1,l−1Dk,lϕ)A1...Ak

A′1...A

′l − k

k + 1�B

(A1ϕA2...Ak )BA′

1...A′l

− k + 2

k + 1�B′(A′

1ϕA1...AkA′

2...A′l )B′ , k � 0, l � 1, (2.7 f )

(Ck−1,l+1C†k,lϕ)A1...Ak

A′1...A

′l = (C †

k+1,l−1Ck,lϕ)A1...AkA′

1...A′l

+(

1

k + 1− 1

l + 1

)(Tk−1,l−1Dk,lϕ)A1...Ak

A′1...A

′l

−�(A1BϕA2...Ak )B

A′1...A

′l + �B′(A′

1ϕA1...AkA′

2...A′l )B′ , k � 1, l � 1. (2.7g)

Proof. We first observe that (2.7a) and (2.7b) are related by complex conjugation. Likewise(2.7c) and (2.7d) as well as (2.7e) and (2.7 f ) are also related by complex conjugation.Furthermore, (2.7g) is given by the difference between (2.7e) and (2.7 f ). It is thereforeenough to prove (2.7a), (2.7d) and (2.7e). We consider each in turn.

• We first prove (2.7a). We partially expand the symmetry, identify the commutator in oneterm, and commute derivatives in the other:

(Dk+1,l−1Ck,lϕ)A1...AkA′

1...A′l−2

= ∇BB′∇(A1C′

ϕA2...AkB)A′

1...A′l−2 B′C′

= 1

k + 1∇B(B′∇B

C′)ϕA1...AkA′

1...A′l−2 B′C′ + k

k+1∇B(B′∇(A1

C′)ϕA2...Ak )BA′

1...A′l−2 B′C′

9


= − 1

k + 1�B′C′

ϕA1...AkA′

1...A′l−2 B′C′ + k

k + 1εB

(A1�B′C′ϕA2...Ak )B

A′1...A

′l−2 B′C′

+ k

k + 1∇(A1

C′∇BB′ϕA2...Ak )B

A′1...A

′l−2 B′C′

= k

k + 1(Ck−1,l−1Dk,lϕ)A1...Ak

A′1...A

′l−2 − �B′C′ϕA1...Ak

A′1...A

′l−2B′C′

.

• To prove (2.7d), we first partially expand the symmetrization over the unprimed indices inthe irreducible decomposition (2.2) and symmetrizing over the primed indices. This gives

∇A1(A′

2ϕA2...AkBA′

3...A′l+2 ) = (Tk,lϕ)A1...AkB

A′2...A

′l+2 − 1

k + 1εA1B(C †

k,lϕ)A2...AkA′

2...A′l+2

−k − 1

k + 1εA1(A2 (C

†k,lϕ)A3...Ak )B

A′2...A

′l+2 . (2.8)

Using the definitions of T and C †, commuting derivatives and using (2.8), we have

(Tk−1,l+1C†k,lϕ)A1...Ak

A′1...A

′l+2 = ∇(A1

(A′1∇|B|A′

2ϕA2...Ak )BA′

3...A′l+2)

= �(A′1A′

2ϕA1...AkA′

3...A′l ) + ∇B(A′

1∇(A1A′

2ϕA2...Ak )BA′

3...A′l+2)

= �(A′1A′

2ϕA1...AkA′

3...A′l ) + ∇B(A′

1 (Tk,lϕ)A1...AkBA′

2...A′l+2)

− 1

k + 1ε(A1|B|∇B(A′

1 (C †k,lϕ)A2...Ak )

A′2...A

′l+2 )

= �(A′1A′

2ϕA1...AkA′

3...A′l ) + (C †

k+1,l+1Tk,lϕ)A1...AkA′

1...A′l+2

+ 1

k + 1(Tk−1,l+1C

†k,lϕ)A1...Ak

A′1...A

′l+2 . (2.9)

Isolating the C †T -term gives (2.7d).• Finally to prove (2.7e), we assume k � 1 and observe


1...A′l

= − ∇BB′∇(B

(B′ϕA1...Ak )

A′1...A

′l )

= − 1

k + 1∇B

B′∇B(B′

ϕA1...AkA′

1...A′l ) − k

k + 1∇B

B′∇(A1(B′

ϕA2...Ak )BA′

1...A′l )

= 1

k + 1(Dk+1,l+1Tk,lϕ)A1...Ak

A′1...A

′l − k

(k + 1)2(Ck−1,l+1C

†k,lϕ)A1...Ak

A′1...A

′l

− k

k + 1∇B

B′∇(A1(B′

ϕA2...Ak )BA′

1...A′l ), (2.10)

where we in the last step used the irreducible decomposition (2.2) on the first term. Wecan solve for the DT -term from which it follows that


1...A′l

= − 1

k + 1(Ck−1,l+1C

†k,lϕ)A1...Ak

A′1...A

′l − ∇B

B′∇(A1(B′

ϕA2...Ak )BA′

1...A′l )

= − 1

k + 1(Ck−1,l+1C

†k,lϕ)A1...Ak

A′1...A

′l − 1

l + 1∇B

B′∇(A1B′ϕA2...Ak )B

A′1...A

′l

− l

l + 1∇B

B′∇(A1(A′

1ϕA2...Ak )BA′

2...A′l )B

′

= − 1

k + 1(Ck−1,l+1C

†k,lϕ)A1...Ak

A′1...A

′l − 1

l + 1∇(A1

B′∇B|B′|ϕA2...Ak )BA′

1...A′l

− 2

l + 1�B

(A1ϕA2...Ak )BA′

1...A′l − l

l + 1∇(A1

(A′1∇|B||B′|ϕA2...Ak )B

A′2...A

′l )B

′

10


− l

l + 1�B

(A1ϕA2...Ak )BA′

1...A′l − l

l + 1�B′(A′

1ϕA1...AkA′

2...A′l )B′

= − (1

k + 1+ 1

l + 1)(Ck−1,l+1C

†k,lϕ)A1...Ak

A′1...A

′l

+ l(l + 2)

(l + 1)2(Tk−1,l−1Dk,lϕ)A1...Ak

A′1...A

′l

− l + 2

l + 1�B

(A1ϕA2...Ak )BA′

1...A′l − l

l + 1�B′(A′

1ϕA1...AkA′

2...A′l )B′ . (2.11)

�

Remark 19. The operators D , C , C † and T together with the irreducible decomposition (2.2)and the relations in lemma 18 have all been implemented in the SymManipulator packageversion 0.9.0 [16].

2.3. Integrability conditions for Killing spinors

Here we demonstrate a procedure for obtaining an integrability condition for a Killing spinorof arbitrary valence. Let κA1...Ak

A′1...A

′l ∈ ker Tk,l . By applying the C operator l + 1 times to the

Killing spinor equation, and repeatedly commute derivatives with (2.7c) we get

0 = (Ck+l+1,1Ck+l,2 · · ·Ck+2,lCk+1,l+1︸︷︷︸l+1

Tk,lκ)A1...Ak+l+2

= l

l + 1(Ck+l+1,1Ck+l,2 · · · Ck+2,lTk+1,l−1Ck,lκ)A1...Ak+l+2 + curvature terms

= 1

l + 1(Ck+l+1,1Tk+l,0Ck+l−1,1 · · · Ck+1,l−1Ck,lκ)A1...Ak+l+2 + curvature terms

= curvature terms. (2.12)

Here, the curvature terms have l−m derivatives of κ and m derivatives of the curvature spinors,where 0 � m � l. The main idea behind this is the observation that the commutator (2.7c)acting on a spinor field without primed indices only gives curvature terms. In the same waywe can use (2.7d) to get

0 = (C †1,k+l+1C

†2,k+l · · · C †

k,l+2C†k+1,l+1︸︷︷︸

k+1

Tk,lκ)A′1...A

′k+l+2

= curvature terms. (2.13)

2.4. Splitting equations into independent parts

In our derivation of necessary conditions for the existence of symmetry operators, it is crucialthat, at each fixed point in spacetime, we can freely choose the values of the Dirac–Weyl andthe Maxwell field and of the symmetric components of any given order of their derivatives.The remaining components of the derivatives to a given order, which involve at least onepair of antisymmetrized indices, can be solved for using the field equations or curvatureconditions. See sections 4.1 and 5.1 for detailed expressions. In the literature, the conditionthat the symmetric components can be freely and independently specified but that no otherparts can be is referred to as the exactness of the set of fields [14, section 5.10]. The symmetriccomponents of the derivatives are exactly those that can be expressed in terms of the operatorT . One can show that, in a globally hyperbolic spacetime, the Dirac–Weyl and Maxwell fieldseach form exact sets. However, it is not necessary for the spacetime to be globally hyperbolicfor this condition to hold. If the spacetime is such that the fields fail to form an exact set, then

11


our methods still give sufficient conditions for the existence of symmetry operators, but theymay no longer be necessary.

The freedom to choose the symmetric components is used in this paper to show thatequations of the type LABA′

(T1,0φ)ABA′ + MAφA = 0 with (C †1,0φ)A′ = 0 forces L(AB)A′ = 0

and MA = 0 because (T1,0φ)ABA′ and φA can be freely and independently specified at a singlepoint. Similar arguments involving derivatives of up to third order are also used.

In several places we will have equations of the form

0 = SABCA′ (T1,0φ)AB

A′TC, (2.14)

where TA and (T1,0φ)ABA′ are free and independent. In particular all linear combinations of theform (T1,0φ)AB

A′TC will then span the space of spinors WABC

A′ = W(AB)CA′

. As the equation(2.14) is linear we therefore get

0 = SABCA′WABC

A′, (2.15)

for all WABCA′ = W(AB)C

A′. We can then make an irreducible decomposition

WABCA′ = W(ABC)

A′ − 23W(A

D|D|A′εB)C, (2.16)

which gives

0 = (− 13 SB

CCA′ − 1

3 SCBCA′

)W BA

AA′ − SABCA′W (ABC)A′

. (2.17)

As WABCA′

is free, its irreducible components W(ABC)A′

and WAD

DA′

are free and independent.We can therefore conclude that

0 = SBC

CA′ + SCBCA′ , (2.18a)

0 = S(ABC)A′ . (2.18b)

Observe that we only get the symmetric part in the last equation due to the symmetry ofW(ABC)

A′.

Instead of introducing a new spinor WABCA′

we will in the rest of the paper work directlywith the irreducible decomposition of (T1,0φ)AB

A′TC and get

0 = (− 13 SA

CCA′ − 1

3 SCACA′

)TB(T1,0φ)ABA′ − SABCA′T (A(T1,0φ)BC)A′

. (2.19)

The formal computations will be the same, and by the argument above, the symmetrizedcoefficients for the irreducible parts TB(T1,0φ)ABA′

and T (A(T1,0φ)BC)A′will individually have

to vanish.

3. The conformal wave equation

For completeness we give here a detailed description of the symmetry operators for theconformal wave equation.

Theorem 20 ([8]). The equation

(� + 4�)φ = 0, (3.1)

has a symmetry operator φ → χ , with order less or equal to two, if and only if there arespinors LAB

A′B′ = L(AB)(A′B′), PAA′ and Q such that

(T2,2L)ABCA′B′C′ = 0, (3.2a)

(T1,1P)ABA′B′ = 0, (3.2b)

(T0,0Q)AA′ = 2

5

(O(0)

2,2L)

AA′. (3.2c)

12


The symmetry operator then takes the form

χ = − 35 LABA′B′

�ABA′B′φ + Qφ + 14φ(D1,1P) + 1

15φ(D1,1D2,2L) + PAA′(T0,0φ)AA′

+ 23 (D2,2L)AA′

(T0,0φ)AA′ + LABA′B′(T1,1T0,0φ)ABA′B′ . (3.3)

The existence of Q satisfying (3.2c) is exactly the auxiliary condition (A0). The proof canalso be carried out using the same technique as in the rest of the paper.

4. The Dirac–Weyl equation

The following theorems imply theorem 4.

Theorem 21. There exists a symmetry operator of the first kind for the Dirac–Weylequation φA → χA, with order less or equal to two, if and only if there are spinor fieldsLAB


(T2,2L)ABCA′B′C′ = 0, (4.1a)

(T1,1P)ABA′B′ = − 1

3 (O(1)

2,2L)ABA′B′

, (4.1b)

(T0,0Q)AA′ = 3

10 (O(0)

2,2L)AA′. (4.1c)


χA = − 815 LBCA′B′

�BCA′B′φA + QφA + 12φB(C1,1P)AB + 2

9φB(C1,1D2,2L)AB + 38φA(D1,1P)

+ 215φA(D1,1D2,2L) + PBA′

(T1,0φ)ABA′ + 89 (D2,2L)BA′

(T1,0φ)ABA′

+ 23 (C2,2L)ABCA′ (T1,0φ)BCA′ + LBCA′B′

(T2,1T1,0φ)ABCA′B′ . (4.2)

Remark 22.

(i) Observe that (4.1b) is the auxiliary condition (A1) for existence of a symmetry operatorof the first kind for Maxwell equation, and (4.1c) is the auxiliary condition (A0) forexistence of a symmetry operator for the conformal wave equation.

(ii) With LABA′B′ = 0 the first order operator takes the form

χA = LPφA + QφA. (4.3)

Theorem 23. There exists a symmetry operator of the second kind for the Dirac–Weyl equationφA → ωA′ , with order less or equal to two, if and only if there are spinor fields LABC

A′ = L(ABC)A′

and PAB = P(AB) such that

(T3,1L)ABCDA′B′ = 0, (4.4a)

(T2,0P)ABCA′ = 0, (4.4b)

0 = − 98�ABCD(D3,1L)CD + 9

5 L(ACDA′

(C2,2�)B)CDA′

− 52�(A

CDF (C3,1L)B)CDF − 2LCDFA′(T4,0�)ABCDFA′ . (4.4c)

The operator takes the form

ωA′ = − 12 LBCDB′�CD

A′ B′φB + 2

3φB(C †2,0P)BA′ + 1

4φB(C †2,0D3,1L)BA′ + PBC(T1,0φ)BCA′

+ 34 (D3,1L)BC(T1,0φ)BCA′ + 3

4 (C †3,1L)BCA′B′ (T1,0φ)BCB′

+LBCDB′(T2,1T1,0φ)BCDA′B′ . (4.5)

13


Remark 24. The scheme for deriving integrability conditions in section 2.3 can be used toshow that

0 = − 25 L(ABC

A′(C2,2�)DFH)A′ + 3L(AB

LA′(T4,0�)CDFH)LA′ + 5�(ABC

L(C3,1L)DFH)L

+ 34�(ABCD(D3,1L)FH), (4.6)

follows from (4.4a). Despite the superficial similarity of this equation to the condition (4.4c),we conjecture that (4.4c) does not follow from (4.4a).

4.1. Reduction of derivatives of the field

In our notation, the Dirac–Weyl equation ∇AA′φA = 0, takes the form (C †

1,0φ)A′ = 0. We seethat the only remaining irreducible part of ∇A

A′φB is (T1,0φ)AB

A′. By commuting derivatives

we see that all higher order derivatives of φA can be reduced to totally symmetrized derivativesand lower order terms consisting of curvature times lower order symmetrized derivatives.

Together with the Dirac–Weyl equation, the commutators (2.7e), (2.7c), (2.7d) give

(D2,1T1,0φ)A = − 6�φA, (4.7a)

(C2,1T1,0φ)ABC = − �ABCDφD, (4.7b)

(C †2,1T1,0φ)AA′B′ = − �ABA′B′φB. (4.7c)

The higher order derivatives can be computed by using the commutators (2.7e), (2.7c),(2.7d) together with the equations above and the Bianchi system to get

(D3,2T2,1T1,0φ)ABA′ = 5

6φC(C2,2�)ABCA′ + 10

3 �(ACA′B′

(T1,0φ)B)CB′ − 163 φ(A(T0,0�)B)

A′

−12�(T1,0φ)ABA′ + 3

2�ABCD(T1,0φ)CDA′, (4.8a)

(C3,2T2,1T1,0φ)ABCDA′ = �(AB

A′B′(T1,0φ)CD)B′ + 5

2�(ABCF (T1,0φ)D)F

A′

− 110φ(A(C2,2�)BCD)

A′ − 12φF (T4,0�)ABCDF

A′, (4.8b)

(C †3,2T2,1T1,0φ)AB

A′B′C′ = 83�C

(A(A′B′

(T1,0φ)B)CC′) − 2

9φ(A(C †2,2�)B)

A′B′C′

− 23φC(T2,2�)ABC

A′B′C′ − �A′B′C′D′ (T1,0φ)AB

D′. (4.8c)

Using irreducible decompositions and the equations above, one can in a systematic wayreduce any third order derivative of φA in terms of φA, (T1,0φ)AB

A′, (T2,1T1,0φ)ABC

A′B′and

(T3,2T2,1T1,0φ)ABCDA′B′C′

.

4.2. First kind of symmetry operator for the Dirac–Weyl equation

Proof of theorem 21. The general second order differential operator, mapping a Dirac–Weylfield φA to S1,0 is equivalent to φA → χA, where

χA = NABφB + MA

BCA′(T1,0φ)BCA′ + LA

BCDA′B′(T2,1T1,0φ)BCDA′B′ , (4.9)

and

LABCDA′B′ = LA(BCD)

(A′B′), MABCA′ = MA(BC)

A′. (4.10)

Here, we have used the reduction of the derivatives to the T operator as discussed in section 4.1.The symmetries (4.10) comes from the symmetries of (T1,0φ)AB

A′and (T2,1T1,0φ)ABC

A′B′.

14


To be able to make a systematic treatment of the dependence of different components of thecoefficients, we will use the irreducible decompositions

LABCDA′B′ = L

4,2ABCD

A′B′ + 34 L

2,2(BC

A′B′εD)A, (4.11a)

MABCA′ = M

3,1ABC

A′ + 23 M

1,1(B

A′εC)A, (4.11b)

NAB = N2,0

AB − 12 N

0,0εAB, (4.11c)

where

L2,2

ABA′B′ ≡ LC

ABCA′B′

, M1,1

AA′ ≡ MB

ABA′, N

0,0≡ NA

A,

L4,2

ABCDA′B′ ≡ L(ABCD)

A′B′, M

3,1ABC

A′ ≡ M(ABC)A′, N

2,0AB ≡ N(AB).

We use the convention that a spinor with underscripts Tk,l

is a totally symmetric valence (k, l)

spinor. Using these spinors, we can rewrite (4.9) as

χA = − 12 N

0,0φA − N

2,0ABφB − 2

3 M1,1

BA′(T1,0φ)ABA′ − M

3,1ABCA′ (T1,0φ)BCA′

− 34 L

2,2

BCA′B′(T2,1T1,0φ)ABCA′B′ − L

4,2ABCDA′B′ (T2,1T1,0φ)BCDA′B′

. (4.12)

The condition for the operator φA → χA to be a symmetry operator is

(C †1,0χ)A′ = 0. (4.13)

The definition of the C † operator, the Leibniz rule for the covariant derivative, and theirreducible decomposition (2.2) allows us to write this equation in terms of the fundamentaloperators acting on the coefficients and the field. Furthermore, using the results from theprevious subsection, we see that this equation can be reduced to a linear combination ofthe spinors (T3,2T2,1T1,0φ)ABCD

A′B′C′, (T2,1T1,0φ)ABC

A′B′, (T1,0φ)AB

A′and φA. For a general

Dirac–Weyl field and an arbitrary point on the manifold, there are no relations betweenthese spinors. Hence, they are independent, and therefore their coefficients have to vanishindividually. After the reduction of the derivatives of the field to the T operator, we cantherefore study the different order derivatives in (4.13) separately. We begin with the highestorder, and work our way down to order zero. �

4.2.1. Third order part. The third order derivative term of (4.13) is

0 = − L4,2

ABCDB′C′(T3,2T2,1T1,0φ)ABCDA′B′C′ . (4.14)

We will now use the argument from section 2.4 to derive equations for the coefficients in asystematic way. To get rid of the free index in equation (4.14) we multiply with an arbitraryspinor field T A′

to get

0 = − L4,2

ABCDB′C′T A′

(T3,2T2,1T1,0φ)ABCDA′B′C′ . (4.15)

From the argument in section 2.4 and the observation that T A′(T3,2T2,1T1,0φ)ABCDA′B′C′ is

irreducible we conclude that

L4,2

ABCDA′B′ = 0. (4.16)

15


4.2.2. Second order part. The second order derivative terms of (4.13) can now bereduced to

0 = − M3,1

ABCB′(T2,1T1,0φ)ABCA′B′ + 1

2 (C2,2 L2,2

)ABCB′(T2,1T1,0φ)ABCA′B′

+ 34 (T2,2 L

2,2)ABCA′B′C′ (T2,1T1,0φ)ABCB′C′

. (4.17)

Here we again multiply with an arbitrary spinor field T A′, but here (T2,1T1,0φ)ABCB′C′

T A′is

not irreducible. Therefore, we decompose it into irreducible parts and get

0 = 34 T (A′

(T2,1T1,0φ)|ABC|B′C′)(T2,2 L2,2

)ABCA′B′C′

+(12 (C2,2 L

2,2)ABCB′ − M

3,1

ABCB′)T A′

(T2,1T1,0φ)ABCA′B′ . (4.18)

The argument in section 2.4 tells that the coefficients of the different irreducible parts have tovanish individually which gives

(T2,2 L2,2

)ABCA′B′C′ = 0, (4.19a)

M3,1

ABCA′ = 1

2 (C2,2 L2,2

)ABCA′. (4.19b)

4.2.3. First order part. The first order derivative terms of (4.13) are

0 = − N2,0

AB(T1,0φ)ABA′ + 13 (C1,1M

1,1)AB(T1,0φ)ABA′ − 1

2 (D3,1M3,1

)AB(T1,0φ)ABA′

− 23 L

2,2A

CB′C′�BCB′C′ (T1,0φ)AB

A′ − 6� L2,2

ABA′B′ (T1,0φ)ABB′

+ 43 L

2,2A

CB′C

′�BCA′C′ (T1,0φ)ABB′ + 5

3 L2,2

AC

A′C′�BCB′C′ (T1,0φ)ABB′

+ 34 L

2,2

CDA′B′�ABCD(T1,0φ)ABB′ + 3

4 L2,2

ABC′D′

�A′B′C′D′ (T1,0φ)ABB′

−(C †3,1M

3,1)ABA′B′ (T1,0φ)ABB′ + 2

3 (T1,1M1,1

)ABA′B′ (T1,0φ)ABB′. (4.20)

Here we again multiply with an arbitrary spinor field T A′and decompose (T1,0φ)ABB′

T A′into

irreducible parts. Due to the argument in section 2.4 the coefficients of the different irreducibleparts have to vanish individually which gives

0 = − N2,0

AB + 13 (C1,1M

1,1)AB − 1

4 (D3,1C2,2 L2,2

)AB − 12 L

2,2(A

CA′B′�B)CA′B′ , (4.21a)

0 = − 6� L2,2

ABA′B′ + 3

4 L2,2

CDA′B′�ABCD + 3

4 L2,2

ABC′D′

�A′B′C′D′ − 1

2 (C †3,1C2,2 L

2,2)AB

A′B′

+3 L2,2

(AC(A′ |C′|�B)C

B′)C′ + 2

3 (T1,1M1,1

)ABA′B′

. (4.21b)

Using the commutators (2.7a) and (2.7 f ) together with (4.19a), this reduces to

N2,0

AB = 13 (C1,1M

1,1)AB − 1

6 (C1,1D2,2 L2,2

)AB, (4.22a)

(T1,1M1,1

)ABA′B′ = − 3

8 L2,2

CDA′B′�ABCD + 3

8 L2,2

ABC′D′

�A′B′C′D′ + (T1,1D2,2 L

2,2)AB

A′B′. (4.22b)

Isolating the T terms in (4.22b) leads us to make the ansatz

M1,1

AA′ = − 3

2 PAA′ + (D2,2 L

2,2)A

A′, (4.23)

16


where PAA′

is undetermined. With this ansatz, the first order equations reduce to

(T1,1P)ABA′B′ = 1

4 L2,2

CDA′B′�ABCD − 1

4 L2,2

ABC′D′

�A′B′C′D′

= 14 (O(1)

2,2 L2,2

)ABA′B′

, (4.24a)

N2,0

AB = − 12 (C1,1P)AB + 1

6 (C1,1D2,2 L2,2

)AB. (4.24b)

4.2.4. Zeroth order part. Using the equations above, the zeroth order derivative terms of(4.13) are

0 = φA

(−2�M

1,1AA′ + 2

3 M1,1

BB′�ABA′B′ + 1

4�ABCD(C2,2 L2,2

)BCDA′ − 5

12 L2,2

BCA′ B

′(C2,2�)ABCB′

−(C †2,0 N

2,0)AA′ − 1

6 L2,2

ABB′C′

(C †2,2�)BA′B′C′ − 8

3 L2,2

ABA′B′ (T0,0�)BB′ + 12 (T0,0 N

0,0)AA′

+ 12 L

2,2

BCB′C′(T2,2�)ABCA′B′C′

). (4.25)

Here, there is no reason to multiply with an arbitrary T A′and do an irreducible decomposition

of T A′φA because T A′

φA is already irreducible. Still the argument in section 2.4 gives that thecoefficient of φA will have to vanish. With the substitutions (4.24b) and (4.23), the vanishingof this coefficient is equivalent to

(T0,0 N0,0

)AA′ = − 6�PA

A′ + 2�ABA′

B′PBB′ − 12�ABCD(C2,2 L

2,2)BCDA′

+ 56 L

2,2

BCA′B′(C2,2�)ABCB′ + 1

3 L2,2

ABB′C′

(C †2,2�)B

A′B′C′ − (C †

2,0C1,1P)AA′

+ 13 (C †

2,0C1,1D2,2 L2,2

)AA′ + 4�(D2,2 L

2,2)A

A′ − 43�AB

A′B′ (D2,2 L

2,2)BB′

+ 163 L

2,2AB

A′B′ (T0,0�)BB′ − L

2,2

BCB′C′(T2,2�)ABC

A′B′C′ . (4.26)

To simplify the C †C D term, we first commute the innermost operators with (2.7a). Thenthe outermost operators are commuted with (2.7b). After that, we are left with the operatorDC †C , which can be turned into DT D by using (2.7 f ) and (4.19a). Finally, the DT Doperator can be turned into C †C D and T DD , again by using (2.7 f ), but this time on theoutermost operators. In detail

(C †2,0C1,1D2,2 L

2,2)AA′ = − 3

2∇BA′�B′C′ L2,2

ABB′C′ + 3

2 (C †2,0D3,1C2,2 L

2,2)AA′

= 3�BC(C2,2 L2,2

)ABC

A′ − 32∇BA′�B′C′ L

2,2A

BB′C′ + 3(D2,2C†3,1C2,2 L

2,2)AA′

= 3�BC(C2,2 L2,2

)ABC

A′ − 32∇BA′�B′C′ L

2,2A

BB′C′

− 6∇BC′�(A′ B′L

2,2|AB|C′)B′ − 3∇CB′�(A

B L2,2

C)BA′B′

+ 4(D2,2T1,1D2,2 L2,2

)AA′

= 2�AB(D2,2 L2,2

)BA′ + 6�A′B′ (D2,2 L

2,2)A

B′ + 3�BC(C2,2 L2,2

)ABC

A′

− 32∇BA′�B′C′ L

2,2A

BB′C′ − 6∇BC′�(A′ B′L

2,2|AB|C′)B′

−3∇CB′�(AB L

2,2C)BA′B′ −4(C †

2,0C1,1D2,2 L2,2

)AA′ + 3(T0,0D1,1D2,2 L2,2

)AA′ .

17


Isolating the C †C D terms, expanding the commutators and using (4.19a) yield

(C †2,0C1,1D2,2 L

2,2)AA′ = − 8

5�BCA′ B

′(C2,2 L

2,2)ABCB′ + 6

5�ABCD(C2,2 L2,2

)BCDA′

−2 L2,2

BCA′ B

′(C2,2�)ABCB′ + 6

5 �A′B′C′D′ (C †2,2 L

2,2)A

B′C′D′

− 85�A

BB′C′(C †

2,2 L2,2

)BA′B′C′ − 2 L2,2

ABB′C′

(C †2,2�)BA′B′C′

−12�(D2,2 L2,2

)AA′ + 4415�ABA′B′ (D2,2 L

2,2)BB′ − 64

5 L2,2

ABA′B′ (T0,0�)BB′

+ 35 L

2,2

BCB′C′(T2,2�)ABCA′B′C′ + 3

5 (T0,0D1,1D2,2 L2,2

)AA′ . (4.27)

Using this in (4.26), and using (2.7e) combined with (4.24a) gives

(T0,0 N0,0

)AA′ = − 8

15�BCA′B′(C2,2 L

2,2)ABCB′ + 3

20�ABCD(C2,2 L2,2

)BCDA′

− 112 L

2,2


20 �A′B′C′D′ (C †

2,2 L2,2

)AB′C′D′

− 815�A

BB′C′(C †

2,2 L2,2

)BA′

B′C′ − 112 L

2,2A

BB′C′(C †

2,2�)BA′

B′C′

− 1645�AB

A′B′ (D2,2 L

2,2)BB′ + 16

15 L2,2

ABA′

B′ (T0,0�)BB′

− 45 L

2,2

BCB′C′(T2,2�)ABC

A′B′C′ − 3

4 (T0,0D1,1P)AA′ + 1

5 (T0,0D1,1D2,2 L2,2

)AA′.

(4.28)

To simplify the remaining terms, we define

ϒ ≡ L2,2

ABA′B′�ABA′B′. (4.29)

Using (4.19a) the gradient of ϒ reduces to

(T0,0ϒ)AA′ = − 43 L

2,2A

BA′ B

′(T0,0�)BB′ + 2

3�BCA′ B

′(C2,2 L

2,2)ABCB′

+ 23 L

2,2

BCA′ B

′(C2,2�)ABCB′ + 2

3�ABB′C′

(C †2,2 L

2,2)BA′B′C′

+ 23 L

2,2A

BB′C′(C †

2,2�)BA′B′C′ + 49�ABA′B′ (D2,2 L

2,2)BB′

+ L2,2

BCB′C′(T2,2�)ABCA′B′C′ . (4.30)

This can be used to eliminate most of the terms in (4.28). Together with the definition of theoperator O(0)

2,2, we find that (4.28) reduces to

(T0,0 N0,0

)AA′ = 9

20

(O(0)

2,2 L2,2

)A

A′ − 45 (T0,0ϒ)A

A′ − 34 (T0,0D1,1P)A

A′ + 15 (T0,0D1,1D2,2 L

2,2)A

A′.

(4.31)

It is now clear that the ansatz

N0,0

= − 2Q − 45ϒ − 3

4 (D1,1P) + 15 (D1,1D2,2 L

2,2), (4.32)

with Q undetermined gives

(T0,0Q)AA′ = − 9

40

(O(0)

2,2 L2,2

)A

A′. (4.33)

We can now conclude that the only restrictive equations are (4.19a), (4.24a) and (4.33).The other equations give expressions for the remaining coefficients in terms of L

2,2AB

A′B′, PAA′ ,

and Q. For convenience we make the replacement L2,2

ABA′B′ → − 4

3 LABA′B′

.

18


4.3. Second kind of symmetry operator for the Dirac–Weyl equation

Proof of theorem 23. The general second order differential operator, mapping a Dirac–Weylfield φA to S0,1 is equivalent to φA → ωA′ , where

ωA′ = NBA′φB + MA′ BCB′

(T1,0φ)BCB′ + LA′ BCDB′C′(T2,1T1,0φ)BCDB′C′ , (4.34)

where

LA′ABCB′C′ = LA′(ABC)(B′C′), MA′ABB′ = MA′(AB)B′ . (4.35)

Here, we have used the reduction of the derivatives to the T operator as discussed above. Thesymmetries (4.35) comes from the symmetries of (T1,0φ)AB

A′and (T2,1T1,0φ)ABC

A′B′. As we

did above, we will decompose the coefficients into irreducible parts to more clearly see whichparts are independent. The irreducible decompositions of LA′

ABCB′C′

and MA′AB

B′are

LA′ABC

B′C′ = L3,3

ABCA′B′C′ + 2

3 L3,1

ABC(B′

εC′)A′, (4.36)

MA′AB

B′ = M2,2

ABA′B′ − 1

2 M2,0

ABεA′B′, (4.37)

where

L3,1

ABCA′ ≡ LB′

ABCA′

B′ , M2,0

AB ≡ MA′ABA′ ,

L3,3

ABCA′B′C′ ≡ L(A′

ABCB′C′), M

2,2AB

A′B′ ≡ M(A′AB

B′ ).

With these irreducible decompositions, we get

ωA′ = NBA′φB − 1

2 M2,0

BC(T1,0φ)BCA′ − M2,2

BCA′B′ (T1,0φ)BCB′ − 23 L

3,1

BCDB′(T2,1T1,0φ)BCDA′B′

− L3,3

BCDA′B′C′ (T2,1T1,0φ)BCDB′C′. (4.38)

The condition for the operator φA → ωA′ to be a symmetry operator is

(C0,1ω)A = 0. (4.39)

Using the results from section 4.1, we see that this equation can be reduced to a linearcombination of the spinors φA, (T1,0φ)AB

A′, (T2,1T1,0φ)ABC

A′B′and (T3,2T2,1T1,0φ)ABCD

A′B′C′.

As above, we can treat these as independent, and therefore their coefficients have to vanishindividually. After the reduction of the derivatives of the field to the T operator, we cantherefore study the different order derivatives in (4.39) separately. We begin with the highestorder, and work our way down to order zero. �

4.3.1. Third order part. The third order part of (4.39) is

0 = − L3,3

BCDA′B′C′(T3,2T2,1T1,0φ)ABCDA′B′C′ . (4.40)

Using the argument from section 2.4, we see that this implies

L3,3

ABCA′B′C′ = 0. (4.41)

19


4.3.2. Second order part. The second order part of (4.39) now takes the form

0 = − M2,2

BCA′B′(T2,1T1,0φ)ABCA′B′ + 1

2 (C †3,1 L

3,1)BCA′B′

(T2,1T1,0φ)ABCA′B′

+ 23 (T3,1 L

3,1)ABCDA′B′ (T2,1T1,0φ)BCDA′B′

. (4.42)

Here we multiply with an arbitrary spinor field T A and decompose (T2,1T1,0φ)BCDC′D′T A into


(T3,1 L3,1

)ABCDA′B′ = 0, (4.43a)

M2,2

ABA′B′ = 1

2 (C †3,1 L

3,1)AB

A′B′. (4.43b)

4.3.3. First order part. The first order part of (4.39) can now be reduced to

0 = − NBA′(T1,0φ)ABA′ + 1

3 (C †2,0M

2,0)BA′

(T1,0φ)ABA′ − 23 (D2,2M

2,2)BA′

(T1,0φ)ABA′

+ 13 L

3,1BCDB′�CD

A′ B′(T1,0φ)A

BA′ − 512 L

3,1

CDFA′�BCDF (T1,0φ)A

BA′

− 6� L3,1

ABCA′ (T1,0φ)BCA′ + 13 L

3,1BCDB′�A

DA′ B

′(T1,0φ)BCA′

+ 53 L

3,1ACDB′�B

DA′ B

′(T1,0φ)BCA′ + 5

4 L3,1

BDF

A′�ACDF (T1,0φ)BCA′

+ 34 L

3,1A

DFA′�BCDF (T1,0φ)BCA′ − (C2,2M

2,2)ABCA′ (T1,0φ)BCA′

+ 12 (T2,0M

2,0)ABCA′ (T1,0φ)BCA′

. (4.44)

Here we again multiply with an arbitrary spinor field T A and decompose (T1,0φ)BCC′T A into


0 = − NAA′ − 1

3 L3,1

BCDA′�ABCD + 1

3 (C †2,0M

2,0)A

A′ − 23 (D2,2M

2,2)A

A′, (4.45)

0 = − 6� L3,1

ABCA′ − (C2,2M

2,2)ABC

A′ + 2 L3,1

(BC|DB′ |�A)DA′B′ + 2 L

3,1(A

DFA′�BC)DF

+ 12 (T2,0M

2,0)ABC

A′. (4.46)

By (4.43b), the commutator (2.7e) and (4.43a) these reduce to

NAA′ = − 1

3 L3,1

ABCB′�BCA′B′ + 13 (C †

2,0M2,0

)AA′ − 1

6 (C †2,0D3,1 L

3,1)A

A′, (4.47a)

(T2,0M2,0

)ABCA′ = (T2,0D3,1 L

3,1)ABC

A′. (4.47b)

If we make the ansatz

M2,0

AB = − 2PAB + (D3,1 L3,1

)AB, (4.48)

these equations reduce to

NAA′ = − 1

3 L3,1

ABCB′�BCA′B′ − 23 (C †

2,0P)AA′ + 1

6 (C †2,0D3,1 L

3,1)A

A′, (4.49a)

(T2,0P)ABCA′ = 0. (4.49b)

20


4.3.4. Zeroth order part. The zeroth order part of (4.39) can now be reduced to

0 = − 2�M2,0

ABφB + 12 M

2,0

CD�ABCDφB − φB(C1,1N)AB − 120 L

3,1B

CDA′φB(C2,2�)ACDA′

+ 160 L

3,1

BCDA′φA(C2,2�)BCDA′ − 5

12 L3,1

ACDA′

φB(C2,2�)BCDA′

− 13�B

CA′B′φB(C †

3,1 L3,1

)ACA′B′ − 12φA(D1,1N) − 8

3 L3,1

ABCA′φB(T0,0�)CA′

+ 13 L

3,1

CDFA′φB(T4,0�)ABCDFA′ . (4.50)

Here we again multiply with an arbitrary spinor field T A and decompose φBT A into irreducibleparts. Due to the argument in section 2.4 the coefficients of the different irreducible parts haveto vanish individually which gives

0 = − 16 L

3,1

ABCA′(C2,2�)ABCA′ + 1

6�ABA′B′(C †

3,1 L3,1

)ABA′B′ − 12 (D1,1N), (4.51a)

0 = − 2�M2,0

AB + 12 M

2,0

CD�ABCD − (C1,1N)AB − 715 L

3,1(A

CDA′(C2,2�)B)CDA′

− 13�(A

CA′B′(C †

3,1 L3,1

)B)CA′B′ − 83 L

3,1ABCA′ (T0,0�)CA′

+ 13 L

3,1

CDFA′(T4,0�)ABCDFA′ . (4.51b)

The equation (4.49a) together with the commutator (2.7b) gives (4.51a). If we substitute(4.49a) in (4.51b), we get a term with the third order operator C C †D . To handle this weuse the same technique as in section 4.2.4. We first commute the innermost operators with(2.7b). Then the outermost operators are commuted with (2.7a). After that, we are left withthe operator DC C †, which can be turned into DT D by using (2.7e) and (4.19a). Finally, theDT D operator can be turned into C C †D and T DD , again by using (2.7e), but this time onthe outermost operators:

(C1,1C†2,0D3,1 L

3,1)AB = 2(C1,1D2,2C

†3,1 L

3,1)AB + 2∇(A

A′�CD L3,1

B)CDA′

= 3�A′B′ (C †3,1 L

3,1)AB

A′B′ + 3(D3,1C2,2C†3,1 L

3,1)AB + 2∇(A

A′�CD L3,1

B)CDA′

= 3�A′B′ (C †3,1 L

3,1)AB

A′B′ + 2∇CB′�A′ B′L

3,1AB

CA′ − 6∇DA′�(AC L

3,1BD)CA′

+ 3(D3,1T2,0D3,1 L3,1

)AB + 2∇(AA′�CD L

3,1B)CDA′

= 3�A′B′ (C †3,1 L

3,1)AB

A′B′ + 2∇CB′�A′ B′L

3,1AB

CA′ − 6∇DA′�(AC L

3,1BD)CA′

− 4(C1,1C†2,0D3,1 L

3,1)AB − 6�(A

C(D3,1 L3,1

)B)C + 2∇(AA′�CD L

3,1B)CDA′ .

Isolating the C C †D terms and expanding the commutators and using (4.43a) yield

(C1,1C†2,0D3,1 L

3,1)AB = − �B

CDF (C3,1 L3,1

)ACDF − �ACDF (C3,1 L

3,1)BCDF

− 4225 L

3,1B

CDA′(C2,2�)ACDA′ − 42

25 L3,1

ACDA′

(C2,2�)BCDA′

− 32�B

CA′B′(C †

3,1 L3,1

)ACA′B′ − 32�A

CA′B′(C †

3,1 L3,1

)BCA′B′

− 12�(D3,1 L3,1

)AB + 2110�ABCD(D3,1 L

3,1)CD − 12 L

3,1ABCA′ (T0,0�)CA′

+ 25 L

3,1

CDFA′(T4,0�)ABCDFA′ . (4.52)

21


The equation (4.49a) together with the equation above, the commutator (2.7e) and (4.49b)gives

(C1,1N)AB = 4�PAB − �ABCDPCD − 2�(D3,1 L3,1

)AB + 720�ABCD(D3,1 L

3,1)CD

− 1775 L

3,1(A

CDA′(C2,2�)B)CDA′ − 1

3�(ACA′B′

(C †3,1 L

3,1)B)CA′B′

− 13�(A

CDF (C3,1 L3,1

)B)CDF − 83 L

3,1ABCA′ (T0,0�)CA′

+ 115 L

3,1

CDFA′(T4,0�)ABCDFA′ . (4.53)

Due to this, the equation (4.51b) reduces to the auxiliary condition

0 = 34�ABCD(D3,1 L

3,1)CD − 6

5 L3,1

(ACDA′

(C2,2�)B)CDA′

+ 53�(A

CDF (C3,1 L3,1

)B)CDF + 43 L

3,1

CDFA′(T4,0�)ABCDFA′ . (4.54)

We can conclude that the only restrictive equations are (4.43a), (4.49b) and (4.54). The otherequations express the remaining coefficients in terms of L

3,1ABCA′ and PAB. For convenience we

make the replacement L3,1

ABCA′ → − 3

2 LABCA′

.

5. The Maxwell equation

Theorem 25. There exists a symmetry operator of the first kind φAB → χAB, with order lessor equal to two, if and only if there are spinor fields LAB


(T2,2L)ABCA′B′C′ = 0, (5.1a)

(T1,1P)ABA′B′ = − 2

3 (O(1)

2,2L)ABA′B′

, (5.1b)

(T0,0Q)BA′ = 0. (5.1c)


χAB = QφAB + (C1,1A)AB, (5.2)

where

AAA′ = − PBA′φAB + 1

3φBC(C2,2L)ABCA′ − 49φAB(D2,2L)B

A′ − LBCA′ B

′(T2,0φ)ABCB′ . (5.3)

We also note that

(C †1,1A)A′B′ = 0. (5.4)

Remark 26.

(i) Observe that one can add a gradient of a scalar to the potential AAA′ without changingthe symmetry operator. Hence, adding ∇AA′ (�BCφBC) to AAA′ with an arbitrary field �AB

is possible.(ii) With LABA′B′ = 0, the first order operator takes the form

χAB = LPφAB + QφAB. (5.5)

22


Theorem 27. There exists second order a symmetry operator of the second kind φAB → ωA′B′ ,with order less or equal to two, if and only if there is a spinor field LABCD = L(ABCD) such that

(T4,0L)ABCDFA′ = 0. (5.6)


ωA′B′ = (C †1,1B)A′B′ , (5.7)

where

BAA′ = 35φBC(C †

4,0L)ABCA′ + LABCD(T2,0φ)BCDA′ . (5.8)

We also note that

(C1,1B)AB = 0. (5.9)

Remark 28.

(i) Observe that also here we can add a gradient of a scalar to the potential BAA′ withoutchanging the symmetry operator. Hence, adding ∇AA′ (�BCφBC) to BAA′ with an arbitraryfield �AB is possible.

(ii) Due to the equations (5.4) and (5.9), we can use AAA′ + BAA′ as a potential for both χAB

and ωA′B′ .

5.1. Reduction of derivatives of the field

In our notation, the Maxwell equation ∇AA′φAB = 0, takes the form (C †

2,0φ)A′ = 0. From thiswe see that the only irreducible part of ∇A

A′φBC is (T2,0φ)ABC

A′. By commuting derivatives we

see that all higher order derivatives of φAB can be reduced to totally symmetrized derivativesand lower order terms consisting of curvature times lower order symmetrized derivatives.

Together with the Maxwell equation, the commutators (2.7e), (2.7c), (2.7d) gives

(D3,1T2,0φ)AB = − 8�φAB + 2�ABCDφCD, (5.10a)

(C3,1T2,0φ)ABCD = 2�(ABCFφD)F , (5.10b)

(C †3,1T2,0φ)ABA′B′ = 2�(A

C |A′B′|φB)C. (5.10c)

The higher order derivatives can be computed using the commutators (2.7e), (2.7c), (2.7d)together with the equations above and the Bianchi system to get

(D4,2T3,1T2,0φ)ABCA′ = 92�(A

D|A′|B′(T2,0φ)BC)DB′ + 9

2�(ABDF (T2,0φ)C)DFA′

− 152 φ(AB(T0,0�)C)A′ + 21

10φ(AD(C2,2�)BC)DA′

+ 32φDF (T4,0�)ABCDFA′ − 15�(T2,0φ)ABCA′ , (5.11a)

(C4,2T3,1T2,0φ)ABCDFB′ = �(AB|B′|A′(T2,0φ)CDF )A′ + 4�(ABC

H (T2,0φ)DF )HB′

− 15φ(AB(C2,2�)CDF )B′ − φ(A

H (T4,0�)BCDF )HB′ , (5.11b)

(C †4,2T3,1T2,0φ)ABC

A′B′C′ = 92�D

(A(A′B′

(T2,0φ)BC)DC′) − 1

2φ(AB(C †2,2�)C)

A′B′C′

− 32φ(A

D(T2,2�)BC)DA′B′C′ − �A′B′C′

D′ (T2,0φ)ABCD′

. (5.11c)

These can in a systematic way be used to reduce any derivative up to third order of φAB interms of φAB, (T2,0φ)ABC

A′, (T3,1T2,0φ)ABCD

A′B′and (T4,2T3,1T2,0φ)ABCDF

A′B′C′.

23


5.2. First kind of symmetry operator for the Maxwell equation

Proof of theorem 25. The general second order differential operator, mapping a Maxwell fieldφAB to S2,0 is equivalent to φAB → χAB, where

χAB = NABCDφCD + MABCDFA′ (T2,0φ)CDFA′ + LABCDFHA′B′ (T3,1T2,0φ)CDFHA′B′, (5.12)

and

LABCDFHA′B′ = L(AB)

(CDFH)(A′B′ ), MABCDFA′ = M(AB)

(CDF )A′, NAB

CD = N(AB)(CD).

Here, we have used the reduction of the derivatives to the T operator as discussed in section5.1. The symmetries comes from the symmetries of (T2,0φ)ABC

A′and (T3,1T2,0φ)ABCD

A′B′.

To be able to make a systematic treatment of the dependence of different components of thecoefficients, we will use the irreducible decompositions

LABCDFHA′B′ = L

6,2AB

CDFHA′B′ − 43ε(A

(C L4,2

DFH)B)

A′B′ − 35ε(C

(AεB)D L

2,2

FH)A′B′, (5.13a)

MABCDFA′ = M

5,1AB

CDFA′ − 65ε(A

(CM3,1

DF )B)

A′ − 12ε(C

(AεB)DM

1,1

F )A′, (5.13b)

NABCD = N

4,0AB

CD − ε(A(C N

2,0

D)B) − 1

3 N0,0

ε(A(CεD)

B) (5.13c)

where the different irreducible parts are

L2,2

ABA′B′ ≡ LCD

ABCDA′B′

, M1,1

AA′ ≡ MBC

ABCA′, N

0,0≡ NAB

AB,

L4,2

ABCDA′B′ ≡ L(A

FBCD)F

A′B′, M

3,1ABC

A′ ≡ M(AD

BC)DA′, N

2,0AB ≡ N(A

CB)C,

L6,2

ABCDFHA′B′ ≡ L(ABCDFH)

A′B′, M

5,1ABCDF

A′ ≡ M(ABCDF )A′, N

4,0ABCD ≡ N(ABCD).

Now, we want the operator to be a symmetry operator, which means that

(C †2,0χ)AA′ = 0. (5.14)

Using the results from the previous subsection, we see that this equation can be reducedto a linear combination of the spinors (T2,4T1,3T0,2φ)A′B′C′

ABCDF , (T1,3T0,2φ)A′B′ABCD,

(T0,2φ)A′ABC and φAB. For a general Maxwell field and an arbitrary point on the manifold,

there are no relations between these spinors. Hence, they are independent, and therefore theircoefficients have to vanish individually. After the reduction of the derivatives of the Maxwellfield to the T operator, we can therefore study the different order derivatives of φAB in (5.14)separately. �

5.2.1. Third order part. The third order derivative terms of (5.14) are

0 = 23 L

4,2BCDFB′C′ (T4,2T3,1T2,0φ)A

BCDFA′ B

′C′

+ L6,2

ABCDFHB′C′ (T4,2T3,1T2,0φ)BCDFHA′ B

′C′. (5.15)

We can multiply this with an arbitrary vector field T AA′and split

(T4,2T3,1T2,0φ)ABCDFA′B′C′T H

A′ into irreducible parts. Then we get

0 = L6,2

ABCDFHB′C′T (A|A′|(T4,2T3,1T2,0φ)BCDFH)A′ B

′C′

+ 23 L

4,2BCDFB′C′TAA′ (T4,2T3,1T2,0φ)ABCDFA′B′C′

. (5.16)

24


The argument in section 2.4 gives that the symmetrized coefficients of the irreducibleparts T (A|A′|(T4,2T3,1T2,0φ)BCDFH)

A′ B′C′

and TAA′ (T4,2T3,1T2,0φ)ABCDFA′B′C′must vanish. This

means that (5.15) is equivalent to the system

L6,2

ABCDFHB′C′ = 0, (5.17a)

L4,2

BCDFB′C′ = 0. (5.17b)

The only remaining irreducible component of LABCDFHA′B′

is L2,2

ABA′B′

.

5.2.2. Second order part. If we use everything above we find that the second order part of(5.14) reduces to

0 = 35 M

3,1

BCDB′(T3,1T2,0φ)ABCDA′B′ − 2

5 (C2,2 L2,2

)BCDB′(T3,1T2,0φ)ABCDA′B′

+ 35 (T2,2 L

2,2)BCD

A′ B′C′

(T3,1T2,0φ)ABCDB′C′ + M5,1

ABCDFB′ (T3,1T2,0φ)BCDFA′ B

′.

(5.18)

Again contracting with an arbitrary vector T AA′and splitting (T3,1T2,0φ)ABCDA′B′

T FC′into

irreducible parts we find

0 = M5,1

ABCDFB′T (A|A′ |(T3,1T2,0φ)BCDF )A′ B

′ − 35 T A(A′

(T3,1T2,0φ)A|BCD|B′C′)(T2,2 L

2,2)BCDA′B′C′

+TAA′

(35 M

3,1BCDB′ − 2

5 (C2,2 L2,2

)BCDB′

)(T3,1T2,0φ)ABCDA′B′

. (5.19)

Again using the argument in section 2.4 we find

(T2,2 L2,2

)BCDA′B′C′ = 0, (5.20a)

M5,1

ABCDFB′ = 0, (5.20b)

M3,1

BCDB′ = 23 (C2,2 L

2,2)BCDB′ . (5.20c)

5.2.3. First order part. Now, after contracting the first order part of (5.14) with an arbitrarytensor T A

A′ , splitting (T2,0φ)ABCA′TDB′ into irreducible parts, and using the argument in section2.4, we find that the first order part of (5.14) is equivalent to the system

N2,0

BC = 12 (C1,1M

1,1)BC − 1

2 (D3,1C2,2 L2,2

)BC − 32 L

2,2(B

DB′C′�C)DB′C′ , (5.21a)

N4,0

ABCD = 15 (C3,1C2,2 L

2,2)ABCD + 3

10 L2,2

(ABB′C′

�CD)B′C′ , (5.21b)

(T1,1M1,1

)BCA′B′ = 12� L

2,2BC

A′B′ − 95 L

2,2

DFA′B′�BCDF − 6

5 L2,2

BCC′D′

�A′B′C′D′

+(C †3,1C2,2 L

2,2)BC

A′B′ − 6 L2,2

D(B

C′(A′�C)D

B′ )C′ , (5.21c)

(T3,1C2,2 L2,2

)ABCDA′B′ = 3 L

2,2(AB

C′(A′�CD)

B′)C′ + 3 L

2,2(A

FA′B′�BCD)F . (5.21d)

The commutators (2.7a), (2.7 f ) and (2.7c) applied to L2,2

ABA′B′

yield

(D3,1C2,2 L2,2

)AB = 23 (C1,1D2,2 L

2,2)AB − 2 L

2,2(A

CA′B′�B)CA′B′ , (5.22a)

25


(C †3,1C2,2 L

2,2)AB

A′B′ = − 12� L2,2

ABA′B′ + L

2,2

CDA′B′�ABCD + 2 L

2,2AB

C′D′�A′B′

C′D′

− 32 (D3,3T2,2 L

2,2)AB

A′B′ + 6 L2,2

C(A

C′(A′�B)C

B′)C′

+ 43 (T1,1D2,2 L

2,2)AB

A′B′, (5.22b)

(T3,1C2,2 L2,2

)ABCDA′B′ = 3

2 (C3,3T2,2 L2,2

)ABCDA′B′ + 3 L

2,2(AB

C′(A′�CD)

B′)C′

+3 L2,2

(AFA′B′

�BCD)F . (5.22c)

It is now clear that (5.21d) is a consequence of (5.22c) and (5.20a). The commutators(5.22a) and (5.22b) together with (5.20a) can be used to reduce (5.21a) and (5.21c) to

N2,0

BC = 12 (C1,1M

1,1)BC − 1

3 (C1,1D2,2 L2,2

)BC − 12 L

2,2(B

DB′C′�C)DB′C′ , (5.23a)

(T1,1M1,1

)BCA′B′ = − 45 L

2,2

DFA′B′�BCDF + 4

5 L2,2

BCC′D′

�A′B′C′D′ + 43 (T1,1D2,2 L

2,2)BCA′B′ . (5.23b)

Now, in view of the form of (5.23b) we make the ansatz

M1,1

AA′ = 2PAA′ + 43 (D2,2 L

2,2)AA′ , (5.24)

where PAA′ is a new spinor field. With this choice (5.23a) and (5.23b) reduce to

N2,0

BC = (C1,1P)BC + 13 (C1,1D2,2 L

2,2)BC − 1

2 L2,2

(BDB′C′

�C)DB′C′ , (5.25a)

(T1,1P)BCA′B′ = − 25 L

2,2

DFA′B′�BCDF + 2

5 L2,2

BCC′D′

�A′B′C′D′ . (5.25b)

In conclusion, the third, second and first order parts of (5.14) vanishes if and only if (5.17),(5.20), (5.21b), (5.24), (5.25a) and (5.25b) are satisfied.

5.2.4. Zeroth order part. After making irreducible decompositions of the derivatives, using(5.20a) and contracting the remaining part of (5.14) with an arbitrary tensor T AA′

, splittingTAA′φCD into irreducible parts, and using the argument in section 2.4, we find that the orderzero part of (5.14) is equivalent to the system

0 = 4�PBA′ − 43�BCA′B′PCB′ + 2

9�CDA′ B

′(C2,2 L

2,2)BCDB′ − 8

15�BCDF (C2,2 L2,2

)CDFA′

+ 2645 L

2,2

CDA′ B

′(C2,2�)BCDB′ + 2

9�BCB′C′

(C †2,2 L

2,2)CA′B′C′

+ 245 L

2,2B

CB′C′(C †

2,2�)CA′B′C′ + 23 (C †

2,0C1,1P)BA′ + 29 (C †

2,0C1,1D2,2 L2,2

)BA′

+ 83�(D2,2 L

2,2)BA′ − 20

27�BCA′B′ (D2,2 L2,2

)CB′

+ 289 L

2,2BCA′B′ (T0,0�)CB′ − 1

3 (T0,0 N0,0

)BA′ − 13 L

2,2

CDB′C′(T2,2�)BCDA′B′C′ , (5.26a)

0 = PDA′�ABCD + 2�(C2,2 L

2,2)ABCA′ + 1

5 (C †4,0C3,1C2,2 L

2,2)ABCA′ + 2

3�ABCD(D2,2 L2,2

)DA′

− 3775 L

2,2(A

D|A′|B′(C2,2�)BC)DB′ + 7

30 L2,2

(ABB′C′

(C †2,2�)C)A′B′C′

26


− 53 L

2,2(AB|A′ |B

′(T0,0�)C)B′ + 7

10 L2,2

(ADB′C′

(T2,2�)BC)DA′B′C′ − �(AB|A′ |B′PC)B′

− 1915�(A

D|A′|B′(C2,2 L

2,2)BC)DB′ + 1

6�(ABB′C′

(C †2,2 L

2,2)C)A′B′C′

− 59�(AB|A′|B

′(D2,2 L

2,2)C)B′ − 1

5�(ABDF (C2,2 L

2,2)C)DFA′ + 3

5 L2,2

DFA′ B

′(T4,0�)ABCDFB′

− 12 (T2,0C1,1P)ABCA′ − 1

6 (T2,0C1,1D2,2 L2,2

)ABCA′ . (5.26b)

Applying the commutators repeatedly we have in general that

(C †4,0C3,1C2,2 L

2,2)ABC

A′ = 53�A′

B′ (C2,2 L2,2

)ABCB′ − 4

3 (D4,2T3,1C2,2 L2,2

)ABCA′ −�(A

D(C2,2 L2,2

)BC)DA′

+ 54 (T2,0D3,1C2,2 L

2,2)ABC

A′

= 53�A′

B′ (C2,2 L2,2

)ABCB′ −2(D4,2C3,3T2,2 L

2,2)ABC

A′ −�(AD(C2,2 L

2,2)BC)D

A′

− ∇DB′�(AB L2,2

C)DA′

B′ − ∇DB′�(A|D| L2,2

BC)A′

B′ − 54∇(A

A′�B′C′L

2,2BC)B′C′

+ 56 (T2,0C1,1D2,2 L

2,2)ABC

A′

= − 10�(C2,2 L2,2

)ABCA′ − 2�ABCD(D2,2 L

2,2)DA′

− 2(D4,2C3,3T2,2 L2,2

)ABCA′ + 25

3 L2,2

(ABA′B′

(T0,0�)C)B′

+ 4915 L

2,2(A

DA′B′(C2,2�)BC)DB′ + 5

6 L2,2

(ABB′C′

(C †2,2�)C)

A′B′C′

− 72 L

2,2(A

DB′C′(T2,2�)BC)D

A′B′C′ + 19

3 �(ADA′B′

(C2,2 L2,2

)BC)DB′

− 56�(AB

B′C′(C †

2,2 L2,2

)C)A′

B′C′ + 259 �(AB

A′B′(D2,2 L

2,2)C)B′

+ 72�(A

DB′C′(T2,2 L

2,2)BC)D

A′B′C′ − �(AB

DF (C2,2 L2,2

)C)DFA′

− L2,2

DFA′B′(T4,0�)ABCDFB′ + 5

6 (T2,0C1,1D2,2 L2,2

)ABCA′. (5.27)

With this and (5.20a), the equation (5.26b) reduces to

0 = PDA′�ABCD + 4

15�ABCD(D2,2 L2,2

)DA′ + 4

25 L2,2

(AD|A′|B

′(C2,2�)BC)DB′

+ 25 L

2,2(AB

B′C′(C †

2,2�)C)A′B′C′ − �(AB|A′ |B′PC)B′ − 2

5�(ABDF (C2,2 L

2,2)C)DFA′

+ 25 L

2,2

DFA′ B

′(T4,0�)ABCDFB′ − 1

2 (T2,0C1,1P)ABCA′ . (5.28)

Using the commutator

(T2,0C1,1P)ABCA′ = 2PDA′�ABCD + 2(C2,2T1,1P)ABCA′ − 2�(AB|A′ |B

′PC)B′ (5.29)

this becomes

0 = − (C2,2T1,1P)ABCA′ + 415�ABCD(D2,2 L

2,2)D

A′ + 425 L

2,2(A

D|A′|B′(C2,2�)BC)DB′

+ 25 L

2,2(AB

B′C′(C †

2,2�)C)A′B′C′ − 25�(AB

DF (C2,2 L2,2

)C)DFA′ + 25 L

2,2

DFA′ B

′(T4,0�)ABCDFB′ .

(5.30)

However, after substituting (5.25b) in this equation, decomposing the derivatives intoirreducible parts and using (5.20a), this equation actually becomes trivial.

27


Doing the same calculations as for the Dirac–Weyl case we see that (4.27) also holds forthe Maxwell case. Directly from the commutators we find

(C †2,0C1,1P)AA′ = − 6�PAA′ + 2�ABA′B′PBB′ − (D2,2T1,1P)AA′ + 3

4 (T0,0D1,1P)AA′ . (5.31)

With this, (4.27) and (5.20a) we can reduce (5.26a) to

0 = − 215�CD

A′ B′(C2,2 L

2,2)BCDB′ − 4

15�BCDF (C2,2 L2,2

)CDFA′ + 2

15 L2,2

CDA′ B

′(C2,2�)BCDB′

+ 415 �A′B′C′D′ (C †

2,2 L2,2

)BB′C′D′ − 2

15�BCB′C′

(C †2,2 L

2,2)CA′B′C′

− 25 L

2,2B

CB′C′(C †

2,2�)CA′B′C′ − 445�BCA′B′ (D2,2 L

2,2)CB′

− 23 (D2,2T1,1P)BA′ + 4

15 L2,2

BCA′B′ (T0,0�)CB′ − 13 (T0,0 N

0,0)BA′

− 15 L

2,2

CDB′C′(T2,2�)BCDA′B′C′ + 1

2 (T0,0D1,1P)BA′ + 215 (T0,0D1,1D2,2 L

2,2)BA′ . (5.32)

Using (5.25b) and the irreducible decompositions, we find

(D2,2T1,1P)BA′ = − 25�BCDF (C2,2 L

2,2)CDF

A′ + 25 L

2,2

CDA′ B

′(C2,2�)BCDB′

+ 25 �A′B′C′D′ (C †

2,2 L2,2

)BB′C′D′ − 2

5 L2,2

BCB′C′

(C †2,2�)CA′B′C′ . (5.33)

To simplify the remaining terms, we use the same trick as for the Dirac–Weyl case. Thedefinition (4.29) and the equation (4.30) can be used together with (5.33), to reduce theequation (5.26b) to

0 = − 13 (T0,0 N

0,0)BA′ − 1

5 (T0,0ϒ)BA′ + 12 (T0,0D1,1P)BA′ + 2

15 (T0,0D1,1D2,2 L2,2

)BA′ . (5.34)

We therefore make the ansatz

N0,0

= 3Q − 35ϒ + 3

2 (D1,1P) + 25 (D1,1D2,2 L

2,2). (5.35)

Now, (5.34) becomes

0 = (T0,0Q)AA′ . (5.36)

5.2.5. Potential representation. From all this we can conclude that the only equations thatrestrict the geometry are (5.20a) and (5.25b). Now, the operator takes the form

χAB = 13 N

0,0φAB + N

4,0ABCDφCD − N

2,0(A

CφB)C − 45 (C2,2 L

2,2)(A

CDA′(T2,0φ)B)CDA′

+ 12 M

1,1CA′ (T2,0φ)AB

CA′ + 35 L

2,2

CDA′B′(T3,1T2,0φ)ABCDA′B′ (5.37)

where N0,0

, N2,0

AB, N4,0

ABCD, M1,1

AA′ are given by (5.35), (5.25a), (5.21b) and (5.24) respectively.

We can in fact simplify this expression by defining the following spinor

AAA′ ≡ PBA′φAB + 1

5φBC(C2,2 L2,2

)ABCA′ + 415φA

B(D2,2 L2,2

)BA′ + 35 L

2,2BCA′B′ (T2,0φ)A

BCB′. (5.38)

Substituting this onto the following, and comparing with (5.37), we find

(C1,1A)AB = − QφAB + χAB − 115φ(A

C(C1,1D2,2 L2,2

)B)C + 110φ(A

C(D3,1C2,2 L2,2

)B)C

− 110 L

2,2(A

CA′B′�|CD

A′B′|φB)D − 110 L

2,2

CDA′B′�(A|CA′B′|φB)D

= − QφAB + χAB, (5.39)

where the last equality follows from a commutator relation. In fact the coefficients in AAA′

were initially left free, and then chosen so all first and second order derivatives of φAB whereeliminated in (5.39).

28


We also get

(C †1,1A)A′B′ = 12

5 � L2,2

ABA′B′φAB − 35 L

2,2

CDA′B′�ABCDφAB + 1

5φAB(C †3,1C2,2 L

2,2)ABA′B′

− 65 L

2,2

AB(A′C

′�|A|CB′ )C′φBC − φAB(T1,1P)ABA′B′

− 35 (T2,2 L

2,2)ABCA′B′C′ (T2,0φ)ABCC′ − 4

15φAB(T1,1D2,2 L2,2

)ABA′B′

= 0 (5.40)

where we in the last step used (5.25b), a commutator and (5.20a).To get the highest order coefficient equal to 1 in AAA′ and in χAB, we define a new

symmetric spinor, which is just a rescaling of L2,2

ABA′B′

LABA′B′ ≡ 35 L

2,2ABA′B′ . (5.41)

Now, the only equations we have left are

(T2,2L)ABCA′B′C′ = 0, (5.42a)

(T1,1P)ABA′B′ = − 2

3 (O(1)

2,2L)ABA′B′

, (5.42b)

(T0,0Q)BA′ = 0, (5.42c)

AAA′ = PBA′φAB + 1

3φBC(C2,2L)ABCA′ + 49φA

B(D2,2L)BA′ − LBCA′ B

′(T2,0φ)ABCB′ . (5.42d)

5.3. Second kind of symmetry operator for the Maxwell equation

For the symmetry operators of the second kind, one can follow the same procedure as above.However, this case was completely handled in [7]. In that paper it was shown that a symmetryoperator of the second kind always has the form φAB → ωA′B′ ,

ωA′B′ = 35φCD(C †

3,1C†4,0L)CDA′B′ − 8

5 (C †4,0L)CDF

(A′ (T2,0φ)|CDF|B′)

+LCDFH (T3,1T2,0φ)CDFHA′B′ , (5.43)

where LABCD = L(ABCD) satisfies

(T4,0L)ABCDFA′ = 0. (5.44)

Hence, the treatment in [7] is satisfactory. However, it is interesting to see if the operator canbe written in terms of a potential. Let

BAA′ ≡ 35φBC(C †

4,0L)ABCA′ + LABCD(T2,0φ)BCDA′ . (5.45)

Then, from the definition of C †, the irreducible decompositions and (5.44) we get

(C †1,1B)A′B′ = 3

5φAB(C †3,1C

†4,0L)ABA′B′ − 8

5 (C †4,0L)ABC

(A′ (T2,0φ)|ABC|B′ )

+ LABCD(T3,1T2,0φ)ABCDA′B′

= ωA′B′ . (5.46)

The coefficients in (5.45) where initially left free, and then chosen to get (5.46).

29


We also get

(C1,1B)AB = 6�LABCDφCD − 32 LAB

FH�CDFHφCD + 35φCD(C3,1C

†4,0L)ABCD

+ 310φ(A

C(D3,1C†4,0L)B)C − 3

2 L(ACDF�B)CD

HφFH − 12 L(A

CDF�|CDF|HφB)H

− (T4,0L)ABCDFA′ (T2,0φ)CDFA′

= − 12φCD(D5,1T4,0L)ABCD − 4

5φBCL(A

DFH�C)DFH − 45φA

CL(BDFH�C)DFH

− (T4,0L)ABCDFA′ (T2,0φ)CDFA′

= 0. (5.47)

Here, we have used (5.44) together with the irreducible decomposition of LABFH�CDFH and

the relations

(D3,1C†4,0L)AB = − 2L(A

CDF�B)CDF , (5.48a)

(C3,1C†4,0L)ABCD = − 10�LABCD − 5

6 (D5,1T4,0L)ABCD + 5L(ABFH�CD)FH, (5.48b)

L(BDFH�C)DFH = 0. (5.48c)

The last equation follows from the integrability condition (cf section 2.3)

L(ABCL�DFH)L = − 1

4 (C5,1T4,0L)ABCDFH = 0, (5.49)

as explained in [7].

6. Factorizations

In this section we will consider special cases for which the auxiliary conditions will alwayshave a solution. We will now prove proposition 7, considering each case in turn.

6.1. The case when LABA′B′ factors in terms of conformal Killing vectors

Proof of proposition 7 part (i). If ξAA′ and ζAA′ are conformal Killing vectors, i.e.

(T1,1ξ )ABA′B′ = 0, (T1,1ζ )AB

A′B′ = 0, (6.1)

then we have a solution

LξζABA′B′ ≡ ζ(A

(A′ξB)

B′) (6.2)

to the equation

(T2,2Lξζ )ABCA′B′C′ = 0. (6.3)

Let

Qξζ ≡ �ζ AA′ξAA′ + 1

3�ABA′B′ζ AA′ξBB′ + 1

8 (C1,1ζ )AB(C1,1ξ )AB

+ 16ξAA′

(C0,2C†1,1ζ )AA′ + 1

6ζ AA′(C0,2C

†1,1ξ )AA′ + 1

8 (C †1,1ζ )A′B′

(C †1,1ξ )A′B′

− 132 (D1,1ζ )(D1,1ξ ), (6.4a)

PξζAA′ ≡ 14ξB

A′ (C1,1ζ )AB + 14ζ B

A′ (C1,1ξ )AB − 14ξA

B′(C †

1,1ζ )A′B′ − 14ζA

B′(C †

1,1ξ )A′B′ . (6.4b)

30


Applying the T operator to the equation (6.4a), decomposing the derivatives intoirreducible parts and using (6.1) gives a long expression with the operators D , C , C †, C C †,C †C , T D , T C , T C †, DC C †, C C C † and C †C C † operating on ξAA′

and ζ AA′. Using the

commutators (2.7a), (2.7c), (2.7d), (2.7e) and (2.7 f ) on the outermost operators and using(6.1), the list of operators appearing can be reduced to the set D , C , C †, DT C †, C T C †

and C C C †. Then using the relations (2.7d) and (2.7e) on the innermost operators the list ofoperators appearing is reduced to C , C †, C T D , where the latter can be eliminated with (2.7c)on the outer operators. After making an irreducible decomposition of ξAA′

ζ BB′and identifying

the symmetric part though (6.2), one is left with

(T0,0Qξζ )AA′ = Lξζ


4�ABCDξBA′(C1,1ζ )CD + 1

4�ABCDζ BA′(C1,1ξ )CD

+Lξζ ABB′C′

(C †2,2�)B

A′B′C′ + 1

4 �A′B′C′D′ξA

B′(C †

1,1ζ )C′D′

+ 14 �A′

B′C′D′ζAB′

(C †1,1ξ )C′D′

. (6.5)

Applying the T operator to the equation (6.4b), decomposing the derivatives intoirreducible parts and using (6.1) gives a expression with the operators C C †, C †C , T C ,T C † operating on ξAA′

and ζ AA′. Using the commutators (2.7c), (2.7d) and (2.7g) and using

(6.1), the entire expression can be reduced to only contain curvature terms. After making anirreducible decomposition of ξAA′

ζ BB′and identifying the symmetric part though (6.2), one is

left with

(T1,1Pξζ )ABA′B′ = Lξζ

CDA′B′�ABCD − Lξζ AB

C′D′�A′B′

C′D′ . (6.6)

Substituting (6.2) into the definition of O(0)

2,2, allows us to see that (6.5) and (6.6) reduces to

(T0,0Qξζ )AA′ = (

O(0)

2,2Lξζ

)A

A′, (6.7a)

(T1,1Pξζ )ABA′B′ = (

O(1)

2,2Lξζ

)AB

A′B′. (6.7b)

The actual form of (6.4a) and (6.4b) was obtained by making sufficiently generalsymmetric second order bi-linear ansatze. The coefficients where then chosen to eliminateas many extra terms as possible in (6.7a) and (6.7b). �

6.2. The case when LABA′B′ factors in terms of Killing spinors

Another way of constructing conformal Killing tensors is to make a product of valence (2, 0)

and valence (0, 2) Killing spinors. It turns out that also this case admits solutions to theauxiliary conditions.

In principle we could construct LABA′B′ from two different Killing spinors, but if thedimension of the space of Killing spinors is greater than one, the spacetime has to be locallyisometric to Minkowski space. In these spacetimes the picture is much simpler and has beenstudied before. The auxiliary conditions will be trivial in these cases. We will therefore onlyconsider one Killing spinor.

Proof of proposition 7 part (ii). Let κAB be a Killing spinor, i.e. a solution to

(T2,0κ)ABCA′ = 0. (6.8)

We have a solution

LκABA′B′ ≡ κABκA′B′

, (6.9)

to the equation

(T2,2Lκ )ABCA′B′C′ = 0. (6.10)

31


Now, let

Qκ ≡ 23�ABA′B′κABκA′B′ + 1

9κAB(C1,1C0,2κ )AB + 427 (C0,2κ )AA′

(C †2,0κ)AA′

+ 19 κA′B′

(C †1,1C

†2,0κ)A′B′ , (6.11a)

PκAA′ ≡ 43κAB(C0,2κ )B

A′ − 43 κA′B′ (C †

2,0κ)AB′. (6.11b)

Applying the T operator to the equation (6.11a), decomposing the derivatives intoirreducible parts and using (6.8) gives a long expression with the operators C , C †, DC ,DC †, C C †, C †C , T C , T C †, C C †C †, C †C C , T C C and T C †C † operating on κAB andκA′B′ . Using the commutators (2.7a), (2.7b), (2.7c), (2.7d), (2.7e) and (2.7 f ) on the outermostoperators and using (6.8), the list of operators appearing can be reduced to the set C , C †,C T C , C †T C †, DT C and DT C †. Then using the relations (2.7a), (2.7b), (2.7c) and (2.7d)on the innermost operators the expression will only contain the operators C , C †:

(T0,0Qκ )AA′ = κBCκA′B′

(C2,2�)ABCB′ − 29�BC

A′B′κBC(C0,2κ )A

B′ + 13�ABCDκCD(C0,2κ )BA′

+ 29�BC

A′B′κA

C(C0,2κ )BB′ − 29�AC

A′B′κB

C(C0,2κ )BB′

+ κABκB′C′

(C †2,2�)B

A′B′C′ + 1

3 �A′B′C′D′ κC′D′

(C †2,0κ)A

B′

− 29�ABB′C′ κB′C′

(C †2,0κ)BA′ + 2

9�ABB′C′ κA′C′(C †

2,0κ)BB′

− 29�AB

A′C′ κB′C

′(C †

2,0κ)BB′. (6.12)

Applying the T operator to the equation (6.11b), decomposing the derivatives intoirreducible parts and using (6.8) gives an expression with the operators C C †, C †C , T Cand T C † operating on κAB and κA′B′ . Using the commutators (2.7c), (2.7d), (2.7e) and (2.7 f )and using (6.8), the expression reduces to

(T1,1Pκ )ABA′B′ = �ABCDκCDκA′B′ − �A′B′

C′D′κABκC′D′. (6.13)

Substituting (6.9) into the definition of O(0)

2,2, and making an irreducible decomposition ofκAB(C0,2κ )C

B′and κA′B′ (C †

2,0κ)AC′ , allows us to see that (6.12) and (6.13) reduces to

(T0,0Qκ )AA′ = (

O(0)

2,2Lκ

)A

A′, (6.14a)

(T1,1Pκ )ABA′B′ = (

O(1)

2,2Lκ

)AB

A′B′. (6.14b)

�

6.3. Example of a conformal Killing tensor that does not factor

The following shows that the condition (A0) is non-trivial. We also see that (A1) does notimply (A0). Unfortunately, we have not found any example of a valence (1, 1) Killing spinorwhich does not satisfy (A1).

Consider the following Stackel metric (see [9] and [29] for more general examples)

gab = dt2 − dz2 − (x + y)(dx2 + dy2) (6.15)

with the tetrad

la = 1√2(∂t )

a + 1√2(∂z)

a, na = 1√2(∂t )

a − 1√2(∂z)

a,

ma = (∂x)a

√2(x + y)1/2

+ i(∂y)a

√2(x + y)1/2

.

32


Expressed in the corresponding dyad (oA, ιA), the curvature takes the form

�ABCD = − 12�o(AoBιCιD), �ABA′B′ = 12�o(AιB)o(A′ ιB′ ), � = 1

12(x + y)3. (6.16)

We can see that the spinor

LABA′B′ = 1

2 (x + y)(oA′oB′

ιAιB + oAoB ιA′ιB

′) − (x − y)o(AιB)o

(A′ιB

′) (6.17)

is a trace-free conformal Killing tensor. We trivially have solutions to the auxiliary condition(A1) because (

O(1)

2,2L)

ABA′B′ = LCDA′B′


�A′B′C′D′ = 0. (6.18)

If there is a solution to (1.4) we will automatically have (C1,1O(0)

2,2L)AB = 0 becauseC1,1T0,0 = 0. However, with the current LAB

A′B′we get(

C1,1O(0)

2,2L)

AB = 5io(AιB)

(x + y)5. (6.19)

This is non-vanishing, which means that the auxiliary condition (A0) does not admit a solution.This example shows that the conditions (A0) and (A1) are not equivalent. From the previoustwo sections, we can also conclude that this LAB

A′B′can-not be written as a linear combination

of conformal Killing tensors of the form ζ(A(A′

ξB)B′) or κABκA′B′ . For the more general metric

in [29] we can in fact also construct a valence (2, 2) Killing spinor which trivially satisfiescondition (A1), but which in general will not satisfy condition (A0). It is interesting to notethat in general this metric does not admit Killing vectors, but we can still construct symmetryoperators for the Maxwell equation.

6.4. Auxiliary condition for a symmetry operator of the second kind for the Dirac–Weylequation

Proof of proposition 7 part (iii). Let κAB be a Killing spinor, and ξAA′a conformal Killing

vector, i.e.

(T2,0κ)ABCA′ = 0, (T1,1ξ )ABA′B′ = 0, (6.20)

then we have a solution

LκξABCA′ ≡ κ(ABξC)

A′(6.21)

to the equation

(T3,1Lκξ )ABCDA′B′ = 0. (6.22)

The auxiliary equation (1.8) now takes the form

0 = 34�ABDFκCD(C1,1ξ )C

F + �ABCDξCA′(C †

2,0κ)DA′ − 3

4�ABCDκCD(D1,1ξ )

− 54�(A

CDFκB)C(C1,1ξ )DF − 54�(A

CDFκ|CD|(C1,1ξ )B)F + 65κ(A

CξDA′(C2,2�)B)CDA′

+ 35κCDξ(A

A′(C2,2�)B)CDA′ − 2κDFξCA′

(T4,0�)ABCDFA′ . (6.23)

Using the technique from section 2.3 we get that the integrability conditions for (6.20) are

0 = �(ABCFκD)F , (6.24a)

0 = 12�ABCD(D1,1ξ ) + 2�(ABC

F (C1,1ξ )D)F − 45ξ(A

A′(C2,2�)BCD)A′ + ξFA′

(T4,0�)ABCDFA′ .

(6.24b)

33


Applying the operator C † on the condition (6.24a) gives

0 = − 12�ABCD(C †

2,0κ)DA′ − 9

10κ(AD(C2,2�)BC)DA′ + 1

4κDF (T4,0�)ABCDFA′ . (6.25)

Using (6.24b) to eliminate �ABCD(D1,1ξ ) and (6.25) to eliminate κDF (T4,0�)ABCDFA′ , anddoing an irreducible decomposition of �ABCFκD

F we see that (6.23) reduces to

0 = − 2(C1,1ξ )CD�(ABCFκD)F , (6.26)

which is trivially satisfied due to (6.24a). �

6.5. Factorization of valence (4, 0) Killing spinors with aligned matter

Proof of theorem 8. Assume that the matter field and the curvature are aligned, that is

0 = �(ABCF�D)FA′B′ . (6.27)

Furthermore, assume that �ABCD does not vanish, and assume that there is a solution LABCD to

(T4,0L)ABCDEA′ = 0. (6.28)

The integrability condition (5.49) for this equation together with the non-vanishing of theWeyl spinor, gives that LABCD and �ABCD are proportional (cf [7]). This means that

0 = L(ABCF�D)FA′B′ , (6.29a)

0 = − L(ABCD(T0,0�)F )A′ + L(ABCH (C2,2�)DF )HA′ + 1

5�(AB|A′ |B′(C †

4,0L)CDF )B′ , (6.29b)

where the second equation is obtained by taking a derivative of the first, decomposing thederivatives into irreducible parts, using the Killing spinor equation, and symmetrizing over allunprimed indices.

Split LABCD into principal spinors LABCD = α(AβBγCδD). Now, the Killing spinor equation(6.28), and the alignment equation (6.29a) gives

0 = αAαBαCαDαF (T4,0L)ABCDFA′ = αAβAαBγBαCδCαDαF∇FA′αD, (6.30a)

0 = αAαBαCαDL(ABCF�D)FA′B′ = 1

4αAβAαBγBαCδCαDαF�DFA′B′ . (6.30b)

We will first assume that αA is not a repeated principal spinor of LABCD. This means thatαAβAαBγBαCδC �= 0 and hence αAαB∇A′AαB = 0, that is αA is a shear-free geodesic nullcongruence. We also get αDαF�DFA′B′ = 0. Contracting (6.29b) with αAαBαCαDαF we get

0 = 14αAβAαBγBαCδCαDαFαH (C2,2�)DFHA′ + 1

5�ABA′ B′αAαBαCαDαF (C †

4,0L)CDFB′

= 14αAβAαBγBαCδCαDαFαH (C2,2�)DFHA′ . (6.31)

Hence, αAαBαC(C2,2�)ABCA′ = 0. But the Bianchi equations give

αAαBαC∇DD′�ABCD = αAαBαC(C2,2�)ABCA′ = 0. (6.32)

It follows from the generalized Goldberg–Sachs theorem that αA is a repeated principal spinorof �ABCD, see for instance [15, proposition 7.3.35]. But LABCD and �ABCD are proportional, soαA is a repeated principal spinor of LABCD after all. Without loss of generality, we can assumethat γ A = αA, a relabeling and rescaling of βA, γ A and δA can achieve this. Repeating theargument with βA, we find that also βA is a repeated principal spinor of LABCD. If βAαA = 0,we can repeat the argument again with δA and see that all principal spinors are repeated, i.e.Petrov type N. Otherwise, we have Petrov type D. In conclusion, we have after rescalingLABCD = α(AαBβCβD). Now, let κAB = α(AβB).

34


First assume that αAβA �= 0. Contracting (6.28) with αAαBαCαDβF , αAαBαCβDβF ,αAαBβCβDβF , αAβBβCβDβF we find

0 = αAαBαC(T2,0κ)ABCA′ , 0 = αAαBβC(T2,0κ)ABCA′ ,

0 = αAβBβC(T2,0κ)ABCA′ , 0 = βAβBβC(T2,0κ)ABCA′ .

Hence, (T2,0κ)ABCA′ = 0.If αAβA = 0, we can find a dyad (oA, ιA) so that αA = oA. Then we have LABCD =

υ2oAoBoCoD and κAB = υoAoB. Contracting (6.28) with oAoBoCιDιFυ−1, oAoBιCιDιFυ−1,oAιBιCιDιFυ−1, ιAιBιCιDιFυ−1 we find

0 = oAoBoC(T2,0κ)ABCA′ , 0 = oAoBιC(T2,0κ)ABCA′ ,

0 = oAιBιC(T2,0κ)ABCA′ , 0 = ιAιBιC(T2,0κ)ABCA′ .

Hence, (T2,0κ)ABCA′ = 0.We can therefore conclude that if the curvature satisfies (6.27), �ABCD does not vanish,

and we have a valence (4, 0) Killing spinor LABCD, then we have a valence (2, 0) Killing spinorκAB such that LABCD = κ(ABκCD). �

7. The symmetry operators with factorized Killing spinor

7.1. Symmetry operators for the conformal wave equation

Let us now consider special cases of symmetry operators for the conformal wave equation. Ifwe choose

LABA′B′ = LξζABA′B′ , PAA′ = 0, Q = 25Qξζ . (7.1)

Then the operator takes the form

χ = 12 Lζ Lξφ + 1

2 Lξ Lζ φ. (7.2)

One can also add an arbitrary first order symmetry operator to this.We can also choose

LABA′B′ = LκABA′B′ , PAA′ = 0, Q = 25Qκ . (7.3)

Substituting these expressions into (3.3) gives a symmetry operator, but we have not foundany simpler form than the one given by (3.3).

Remark 29. Apart from factorizations, one can in special cases get symmetry operators fromKilling tensors. If KAB

A′B′is a Killing tensor, then we have

(T2,2L)ABCA′B′C′ = 0, (D2,2L)AA′ = − 34 (T0,0S)AA′ , KABA′B′ = LABA′B′ + 1

4 SεABεA′B′

where LABA′B′ = K(AB)

(A′B′) and S = KAA

A′ A′. The commutator (2.7c) gives (C1,1D2,2L)AB = 0.

If we also assume vacuum, then the equation (4.27) gives

(T0,0D1,1D2,2L)AA′ = − 2�ABCD(C2,2L)BCDA′ − 2�A′B′C′D′ (C †

2,2L)AB′C′D′

. (7.4)

Hence, we can choose

Q = − 115 (D1,1D2,2L), (7.5)

to satisfy condition A0, and get the well known symmetry operator

χ = − 12 (T0,0S)AA′

(T0,0φ)AA′ + LABA′B′(T1,1T0,0φ)ABA′B′ = ∇AA′ (KABA′B′∇BB′φ), (7.6)

which is valid for vacuum spacetimes.

35


7.2. Symmetry operator of the first kind for the Dirac–Weyl equation

Let us now consider special cases of symmetry operators of the first kind for the Dirac–Weylequation. We can choose

LABA′B′ = LξζABA′B′ , PAA′ = − 13Pξζ

AA′, Q = 3

10Qξζ , (7.7)

to get a symmetry operator for the Dirac–Weyl equation. The operator then becomes

χA = 12 Lξ Lζ φA + 1

2 Lζ LξφA. (7.8)

We can add any conformal Killing vector to PAA′and any constant to Q. Note that if we add

the conformal Killing vector 12 (ξBB′∇BB′ζ AA′ − ζ BB′∇BB′ξAA′

) to PAA′, the operator gets the

factored form

χA = Lξ Lζ φA. (7.9)

We can also choose

LABA′B′ = LκABA′B′ , PAA′ = − 13Pκ

AA′, Q = 3

10Qκ . (7.10)

Substituting these expressions into (4.2) gives a symmetry operator, but we have not foundany simpler form than the one given by (4.2).

7.3. Symmetry operator of the first kind for the Maxwell equation

Let us now consider the symmetry operators of the first kind for the Maxwell equation. Let

LABA′B′ = LξζABA′B′ , PAA′ = − 23Pξζ

AA′, Q = 0, (7.11)

to get a symmetry operator. With this choice the symmetry operator and the potential reduceto

χAB = 12 Lζ LξφAB + 1

2 Lξ Lζ φAB, (7.12a)

AAA′ = − 12ζ B

A′LξφAB − 12ξB

A′Lζ φAB. (7.12b)

A general first order operator can be added to this. If we add an the same commutatoras above with an appropriate coefficient to PAA′

, we get the same kind of factorization of theoperator as above.

We can also get a solution by setting

LABA′B′ = LκABA′B′ , PAA′ = − 23Pκ

AA′, Q = 0. (7.13)

With this choice the symmetry operator and the potential reduce to

χAB = (C1,1A)AB, (7.14a)

AAA′ = − 13 AB(C0,2κ )B

A′ + κA′B′ (C †2,0 )A

B′, (7.14b)

AB ≡ − 2κ(ACφB)C. (7.14c)

This proves the first part of theorem 11.

36


7.4. Symmetry operator of the second kind for the Dirac–Weyl equation

Let

LABCA′ = LκξABC

A′, (7.15a)

PAB = − 12 Lξ κAB + 3

8κAB(D1,1ξ )

= 18κAB(D1,1ξ ) − 1

2κ(AC(C1,1ξ )B)C + 1

3ξ(AA′

(C †2,0κ)B)A′ . (7.15b)

Using the equations (6.20), the commutators (2.7e), (2.7 f ), (2.7c), (2.7d) and theirreducible decompositions of �ABCFκD

F and �ABA′B′ξCB′

we get

(T2,0P)ABCA′ = − 16κ(AB(C †

2,0C1,1ξ )C)A′ − 12κ(A

D(T2,0C1,1ξ )BC)DA′ + 18κ(AB(T0,0D1,1ξ )C)A′

+ 16ξ(A|A′|(C1,1C

†2,0κ)BC) + 1

3ξ(AB′

(T1,1C†2,0κ)BC)A′B′

= ξDA′�(ABC

FκD)F

= 0, (7.16)

where we in the last step used the integrability condition (6.24b). Observe that PAB is givenby a conformally weighted Lie derivative, but now with a different weight. The operator Lξ

has a conformal weight adapted to the weight of the conformally invariant operator C †. Theoperator T is also conformally invariant, but with a different weight. This explains the extraterm in PAB.

The symmetry operator of the second kind for the Dirac–Weyl equation now takes theform

ωA′ = κBC(T1,0Lξφ)BCA′ − 23 LξφB(C †

2,0κ)BA′ . (7.17)

Hence, we can conclude that if LABCA′ factors, then one can choose a corresponding PAB sothat the operator factors as a first order symmetry operator of the first kind followed by a firstorder symmetry operator of the second kind.

7.5. Symmetry operator of the second kind for the Maxwell equation

If we let LABCD = κ(ABκCD) with

(T2,0κ)ABCA′ = 0, (7.18)

then the operator if the second kind now takes the form

ωA′B′ = (C †1,1B)A′B′ , (7.19a)

BAA′ = κAB(C †2,0 )B

A′ + 13 AB(C †

2,0κ)BA′ , (7.19b)

AB ≡ − 2κ(ACφB)C. (7.19c)

This proves the second part of theorem 11.

Acknowledgments

The authors would like to thank Steffen Aksteiner and Lionel Mason for helpful discussions.We are particularly grateful to Lionel Mason for his ideas concerning theorem 8. Furthermorewe would like to thank Niky Kamran and J. P. Michel for helpful comments. LA thanks Shing-Tung Yau for generous hospitality and many interesting discussions on symmetry operatorsand related matters, during a visit to Harvard University, where some initial work on thetopic of this paper was done. This material is based upon work supported by the NationalScience Foundation under Grant no. 0932078 000, while the authors were in residence at theMathematical Sciences Research Institute in Berkeley, California, during the semester of 2013.

37


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Second order symmetry operators